Notes on the value function approach to multiobjective bilevel optimization

Abstract

This paper is concerned with the value function approach to multiobjective bilevel optimization which exploits a lower-level frontier-type mapping in order to replace the hierarchical model of two interdependent multiobjective optimization problems by a single-level multiobjective optimization problem. As a starting point, different value-function-type reformulations are suggested and their relations are discussed. Here, we focus on the situations where the lower-level problem is solved up to efficiency or weak efficiency, and an intermediate solution concept is suggested as well. We study the graph-closedness of the associated efficiency-type and frontier-type mappings. These findings are then used for two purposes. First, we investigate existence results in multiobjective bilevel optimization. Second, for the derivation of necessary optimality conditions via the value function approach, it is inherent to differentiate frontier-type mappings in a generalized way. Here, we are concerned with the computation of upper coderivative estimates for the frontier-type mapping associated with the setting where the lower-level problem is solved up to weak efficiency. We proceed in two ways, relying, on the one hand, on a weak domination property and, on the other hand, on a scalarization approach. Illustrative examples visualize our findings and some flaws in the related literature.

KEYWORDS:

• Bilevel optimization
• existence results
• parametric optimization
• multiobjective optimization
• set-valued and variational analysis

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 49J52
• 49J53
• 90C29
• 90C30

1. Introduction

In this note, we are concerned with multiobjective bilevel optimization problems of type (BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) where $F:{\mathbb{R}}^n\times {\mathbb{R}}^m\to {\mathbb{R}}^p$ is a continuous vector-valued function, $\mathcal{K}\subset {\mathbb{R}}^p$ is a nonempty, closed, convex, pointed cone with nonempty interior, $X\subset {\mathbb{R}}^n$ is a nonempty, closed set, and $\hat{\mathrm{\Psi }}:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^m$ is a suitable solution mapping associated with the parametric multiobjective optimization problem (P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) Above, $f:{\mathbb{R}}^n\times {\mathbb{R}}^m\to {\mathbb{R}}^q$ is a continuous vector-valued function, $\mathcal{C}\subset {\mathbb{R}}^q$ is a nonempty, closed, convex, pointed cone with nonempty interior, and $\mathrm{\Gamma }:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^m$ is a set-valued mapping with a closed graph. In our model problem, $\mathcal{K}$ and $\mathcal{C}$ play the role of ordering cones, i.e. they specify what ‘minimization’ means in (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ) and (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ), up to different notions of optimality/efficiency in multiobjective optimization. Indeed, even for a fixed ordering cone, there exist diverse concepts of efficiency which aim to characterize the valuable points in the feasible set when faced with multiple objective functions. In this paper, we focus our attention on so-called efficiency and weak efficiency. Exemplary, we refer the interested reader to the monographs [1,2] for a detailed introduction to multiobjective optimization and to Section 2.2 where we summarize some essential foundations of this topic.

In the case where $p:=q:=1$ and $\mathcal{K}:=\mathcal{C}:={\mathbb{R}}_{+}$, (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ) reduces to a so-called standard bilevel optimization problem which is used to model situations of interdependent decision making between two parties that cooperate with each other. Let us note that this model is closely related to the so-called optimistic approach to bilevel optimization as shown in [3, Proposition 6.9]. In the literature, (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ) and (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ) are referred to as the upper- and lower-level problem, respectively, and we will stick to this terminology even in the more general situation considered here. Due to numerous underlying applications from, e.g. finance, economics, data as well as natural sciences, and energy management, bilevel optimization can be found among the most active fields of research within the optimization community, see the survey papers [4,5].

For an introduction to bilevel optimization, we refer the interested reader to the monographs [6–9]. We emphasize that bilevel optimization problems are, in general, highly irregular and nonconvex optimization problems which are only implicitly given as a closed-form representation of the solution mapping $\hat{\mathrm{\Psi }}$ associated with (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ) is often not available. This causes the model (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ) to be rather challenging from a theoretical as well as numerical perspective.

Since the set of efficient or weakly efficient points of a multiobjective optimization problem is, usually, not a singleton, the minimization of a scalar function over the set of efficient or weakly efficient points, in order to reduce the number of reasonable and economically utilizable points, is a closely related topic, see e.g. [10–13], and such problems already possess a hierarchical structure. So-called semivectorial bilevel optimization problems, where only $p:=1$ and $\mathcal{K}:={\mathbb{R}}_{+}$ are demanded in (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ), i.e. only the underlying parametric optimization problem (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ) possesses multiple objective functions, provide a much more general model paradigm and have been investigated, e.g. in [14–19]. The even more general situation where the objective functions of both decision makers in (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ) are allowed to be vector-valued has been considered, e.g. in [20–25]. For formal completeness, let us also mention that the setting where $q:=1$ and $\mathcal{C}:={\mathbb{R}}_{+}$ hold while the objective function of (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ) is vector-valued is also reasonable, see e.g. [26–28]. It is well known from the literature that scalarization techniques can be used to transfer multiobjective optimization problems into scalar ones, see e.g. [2, Section 5]. Hence, whenever the lower-level problem (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ) possesses multiple objective functions, it seems to be an obvious idea to replace it with a scalarized counterpart and to treat the scalarization parameters as additional upper-level variables. This procedure, however, comes with a price. It has been shown in the setting of semivectorial bilevel optimization that this reformulation induces additional local minimizers which might be attractive for solution algorithms, see [17]. Even worse, the more detailed study [29] revealed that this transformation also induces additional stationary points while the constraint qualifications needed to tackle the reformulated problem might be stronger than the ones applicable to the original model. These observations are likely to concern multiobjective bilevel optimization problems, too. Hence, one should not tackle (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ) with scalarization approaches for the theoretical and numerical treatment of (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ) in a lightheaded way.

From the viewpoint of applications, the investigation of multiobjective bilevel optimization problems is rather relevant. Let us point the reader to three interesting model problems arising in practice. First, for example, in [28], transportation planning systems are considered in which a system manager wants to determine road tolls for a transportation network by deciding on the charges on each edge of the network. Thereby, the leader wants to minimize the total network cost and to maximize the total revenue as well as the consumers' surplus at the same time. For the lower-level problem, the travelers choose their routes in the network by the traffic volume. This circumstance is modeled as an equilibrium traffic assignment problem in which each user chooses the most convenient path selfishly. Second, [30] is concerned with standalone hybrid renewable energy systems which can reliably meet the energy demand of users in remote areas and reduce the intermittency of different renewable energy sources. At the upper-level stage, policy makers like a government give incentive subsidies to investors to fill in the gap between the market price and the price charged to consumers. Thereby, the decision maker is interested to minimize the negative impact of hybrid renewable energy systems on the environment measured by the emitted annualized carbon dioxide and, conflicting to this, the subsidies borne by the government. The investor's objective for the lower-level problem is to reduce the annualized cost of the energy system project which comprise the annualized capital cost, annual fuel cost, annual system maintenance cost, and subsidies. In the constraints, technical components of several energy technologies are considered. Third, in [31], multiobjective optimization problems are used to model waste management with obnoxious effects. The government decides at the upper level about the location and scale of waste recycling centers to minimize the total economic cost and, simultaneously, the abominable effects caused by recycling centers under given capacity constraints. Based on the locations of the recycling centers, a sanitation company determines waste collection routing plans to minimize the vehicle traveling and fixed cost for which, exemplary, flow conservation for delivery, locations of the underlying routing graph, load of the vehicle on its edges, and the recycling center capacities are considered as constraints.

In this note, we are concerned with the so-called value function approach to bilevel optimization, which dates back to [32] in the scalar case. Let us, for a moment, assume that (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ) is a scalar problem, i.e. $q:=1$ and $\mathcal{C}:={\mathbb{R}}_{+}$. With the aid of the so-called lower-level optimal value function $\phi :{\mathbb{R}}^n\to \overline{\mathbb{R}}$ given by (1) $\mathrm{\forall x}\in {\mathbb{R}}^n:\phi \left(x\right):=\underset{y}{inf}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\},$(1) where $\overline{\mathbb{R}}:=\mathbb{R}\cup \{-\mathrm{\infty },\mathrm{\infty }\}$ and $inf\mathrm{\varnothing }:=\mathrm{\infty }$, one can equivalently reformulate (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ) by means of (2) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\le \phi (x),y\in \mathrm{\Gamma }(x\left)\right\},$(2) as the feasible sets of (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ) and (2(2) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\le \phi (x),y\in \mathrm{\Gamma }(x\left)\right\},$(2) ) coincide. Noting that φ is only implicitly available while being inherently nonsmooth, and observing that classical constraint qualifications from nonsmooth programming are likely to fail at all feasible points of (2(2) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\le \phi (x),y\in \mathrm{\Gamma }(x\left)\right\},$(2) ), see [33, Proposition 3.2], (2(2) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\le \phi (x),y\in \mathrm{\Gamma }(x\left)\right\},$(2) ) is still a rather challenging problem. Nevertheless, the so-called value function reformulation (2(2) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\le \phi (x),y\in \mathrm{\Gamma }(x\left)\right\},$(2) ) has turned out to be useful for the construction of necessary optimality conditions and solution algorithms for (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ) in the setting where, additionally, $p:=1$ and $\mathcal{K}:={\mathbb{R}}_{+}$, see e.g. [33–37] and the references therein.

Based on its successful application in the scalar situation, the paper [24] suggests an extension of the value function approach to multiobjective bilevel optimization. In order to recover the approach of [24], let us introduce the set-valued mapping $\hat{\mathrm{\Phi }}:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^q$ by means of $\mathrm{\forall x}\in {\mathbb{R}}^n:\hat{\mathrm{\Phi }}\left(x\right):=f(x,\hat{\mathrm{\Psi }}(x\left)\right):=\left\{f\right(x,y)|y\in \hat{\mathrm{\Psi }}(x\left)\right\},$and observe that (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ) is closely related to (VFR) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \hat{\mathrm{\Phi }}(x),y\in \mathrm{\Gamma }(x\left)\right\}$(VFR) which, again, we will refer to as value function reformulation of (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ). Let us note at this point that (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ) and (VFR(VFR) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \hat{\mathrm{\Phi }}(x),y\in \mathrm{\Gamma }(x\left)\right\}$(VFR) ) do not need to be equivalent anymore in general, see Example 3.19. In [24], the authors exploit the model problem (VFR(VFR) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \hat{\mathrm{\Phi }}(x),y\in \mathrm{\Gamma }(x\left)\right\}$(VFR) ) for the derivation of necessary optimality conditions for the associated model (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ) where the lower-level parametric optimization problem is solved up to efficient points. More precisely, the authors of [24] transfer the classical arguments from [33], which apply to bilevel optimization problems with scalar upper- and lower-level objective functions, to the multiobjective situation. Among others, this makes the (generalized) differentiation of the so-called frontier mapping $\hat{\mathrm{\Phi }}$ from above necessary. Therefore, Mordukhovich's coderivative construction is exploited, see e.g. [38,39]. Let us mention that the computation of coderivatives associated with frontier maps has already been studied in the literature before, see e.g. [40–42].

Now, we would like to mention an important and crucial observation which seemingly has been overlooked in most of the aforementioned literature dealing with multiobjective bilevel optimization. If the lower-level problem (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ) is solved up to efficiency, then the associated solution mapping is likely to possess a graph which is not closed – even under comparatively strict assumptions. This non-closedness also concerns the associated frontier mapping which is used in the definition of the associated value function reformulation. Let us point out two immediate disadvantages which result from this non-closedness. First, it will be difficult to guarantee that (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ) possesses efficient or at least weakly efficient points in this case. Second, for the computation of coderivatives, the underlying set-valued mapping needs to possess, at least locally around the reference point, a closed graph, so, in turn, this concept formally cannot be applied to the frontier mapping of interest in general.

In this note, we tackle both of the aforementioned issues and, along the way, comment on related peculiarities of multiobjective bilevel optimization. Throughout the paper, several examples are presented to illustrate our arguments. Furthermore, we compare our results with related findings applying to (2(2) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\le \phi (x),y\in \mathrm{\Gamma }(x\left)\right\},$(2) ) in the scalar case. To start, let us mention that whenever (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ) is solved up to weak efficiency only, then the associated solution map is likely to possess a closed graph under mild assumptions, and this also extends to the associated frontier mapping. This can be distilled e.g. from [43, Section 3.5], but we provide a self-contained proof here for the purpose of completeness. In Section 3 of the paper, we are not only concerned with some formal properties of the value function reformulation, but also with the derivation of existence results for (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ) which is a closely related issue. As already mentioned, this is reasonable under mild assumptions if (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ) is solved up to weak efficiency. We also show that the efficiency and associated frontier mapping of a fully linear parametric multiobjective optimization problem possess closed graphs, giving rise to yet another criterion for the existence of efficient and weakly efficient points for (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ). Furthermore, we suggest an intermediate solution approach between efficiency and weak efficiency applying to (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ) which naturally comes along with a closed graph of the associated mappings $\hat{\mathrm{\Psi }}$ as well as $\hat{\mathrm{\Phi }}$ and, thus, the superordinate problem (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ) is also likely to possess efficient and weakly efficient points. Section 4 is dedicated to the computation of upper coderivative estimates for the weak frontier mapping, i.e. the frontier mapping associated with (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ) when solved up to weak efficiency. First, we follow the approach promoted in [40], which relies on the so-called domination property, see e.g. [44,45], and point out some associated advantages and disadvantages. Furthermore, we illustrate a crucial error in [24, Theorem 4.1], that is concerned with the computation of an upper estimate for the coderivative of the efficiency-related frontier mapping of (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ). Unluckily, this shows that the results obtained in [24, Section 4] are not reliable. A second approach to coderivative estimates we discuss here is based on a weighted-sum scalarization of the problem (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ) in the convex situation. This idea is motivated by considerations from [19].

The remainder of the paper is organized as follows. In Section 2, we comment on the basic notation used in this paper, as well as some preliminaries from set-valued analysis and generalized differentiation. Furthermore, some foundations of multiobjective optimization are presented. Section 3 is dedicated to the detailed study of model problems in multiobjective bilevel optimization with a particular emphasis on value-function-type reformulations and their relations to each other. Based on a discussion of closedness properties of efficiency-type and associated frontier mappings in Section 3.1, some existence results are stated in Section 3.2. Furthermore, an intermediate way between efficiency and weak efficiency on how to interpret (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ) is suggested in Section 3.3. In Section 4, we are concerned with the derivation of upper coderivative estimates for frontier mappings. Based on a classical approach from the literature, we recover some known results in the novel setting of weak efficiency in Section 4.1. Relying on a scalarization approach, some new findings are obtained in Section 4.2. The paper closes with some concluding remarks in Section 5.

2. Preliminaries

In this section, we comment on the notation used in this paper. Particularly, some foundations of set-valued and variational analysis are provided. Furthermore, we present some necessary basics of multiobjective optimization.

2.1. Basic notation

The notation in this paper is fairly standard and mainly follows [38,39].

Let ${\mathbb{R}}_{+}$ and ${\mathbb{R}}_{-}$ represent the nonnegative and nonpositive real numbers, respectively. Throughout the paper, we equip ${\mathbb{R}}^n$, where $n\in \mathbb{N}$ is a positive integer, with the Euclidean inner product $〈\cdot ,\cdot 〉:{\mathbb{R}}^n\times {\mathbb{R}}^n\to \mathbb{R}$ and the associated Euclidean norm $\Vert \cdot \Vert :{\mathbb{R}}^n\to {\mathbb{R}}_{+}$. We use ${\mathbb{S}}_1\left(0\right)$ to represent the unit sphere around the origin in ${\mathbb{R}}^n$ where the dimension of the underlying space will be always clear from the context. For some set $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $\mathrm{cl}\mathrm{\Omega }$, $\mathrm{int}\mathrm{\Omega }$, $\mathrm{conv}\mathrm{\Omega }$, and $\mathrm{cone}\mathrm{\Omega }$ denote the closure, the interior, the convex hull, and the convex conic hull of Ω, respectively.

Whenever $\upsilon :{\mathbb{R}}^n\to {\mathbb{R}}^m$ is a mapping which is differentiable at $\overline{x}\in {\mathbb{R}}^n$, then ${\upsilon }^{\prime }\left(\overline{x}\right)\in {\mathbb{R}}^{m\times n}$ denotes the Jacobian of υ at $\overline{x}$. Partial derivatives are represented in analogous fashion.

2.1.1. Convexity properties of vector-valued functions

In this paper, we are partially concerned with convexity properties of vector-valued mappings with respect to (w.r.t.) a given so-called ordering cone, see e.g. [2]. For a nonempty, closed, convex, pointed cone $\mathcal{K}\subset {\mathbb{R}}^p$, a mapping $F:{\mathbb{R}}^m\to {\mathbb{R}}^p$ is said to be $\mathcal{K}$-convex if $\mathrm{\forall y},y^{\prime }\in {\mathbb{R}}^m\mathrm{\forall \alpha }\in [0,1]:\mathrm{\alpha F}\left(y\right)+(1-\alpha )F\left(y^{\prime }\right)-F(\mathrm{\alpha y}+(1-\alpha \left)y^{\prime }\right)\in \mathcal{K}.$We note that whenever F is $\mathcal{K}$-convex and $Y\subset {\mathbb{R}}^m$ is convex, then $F\left(Y\right)+\mathcal{K}$ is convex, and the function $y\mapsto 〈\lambda ,F\left(y\right)〉$ is convex for each $\lambda \in {\mathcal{K}}^{\ast }$. Here, we made use of ${\mathcal{K}}^{\ast }:=\{\lambda |\mathrm{\forall z}\in \mathcal{K}:〈\lambda ,z〉\ge 0\}$ to denote the dual cone of $\mathcal{K}$ which is a closed, convex cone as well. Let us mention that a componentwise convex mapping $F:{\mathbb{R}}^m\to {\mathbb{R}}^p$ is ${\mathbb{R}}_{+}^p$-convex.

The mapping F is said to be $\mathcal{K}$-strictly-quasiconvex if $\begin{array}{rl}& \mathrm{\forall y},y^{\prime }\in {\mathbb{R}}^m\mathrm{\forall \alpha }\in (0,1):\\ & F\left(y^{\prime }\right)-F\left(y\right)\in \mathcal{K},y\ne y^{\prime }⟹F\left(y^{\prime }\right)-F(\mathrm{\alpha y}+(1-\alpha \left)y^{\prime }\right)\in \mathrm{int}\mathcal{K}.\end{array}$Exemplary, in the case where $\mathcal{K}:={\mathbb{R}}_{+}^p$, F is ${\mathbb{R}}_{+}^p$-strictly-quasiconvex if $F_1,\dots ,F_p$ are so-called semi-strictly quasiconvex in the sense that $\begin{array}{rl}& \mathrm{\forall y},y^{\prime }\in {\mathbb{R}}^m\mathrm{\forall \alpha }\in (0,1):\\ & F_i\left(y^{\prime }\right)\ge F_i\left(y\right),y\ne y^{\prime }⟹F_i\left(y^{\prime }\right)>F_i(\mathrm{\alpha y}+(1-\alpha \left)y^{\prime }\right)\end{array}$has to hold for all $i=1,\dots ,p$. The latter is clearly inherent whenever the functions $F_1,\dots ,F_p$ are strictly convex.

2.1.2. Set-valued mappings

In this paper, we are concerned with so-called set-valued mappings which assign to each element of the pre-image space a (potentially empty) subset of the image space. Such objects will be denoted by $\mathrm{{\rm Y}}:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^m$. Throughout the paper, we use $\mathrm{gph}\mathrm{{\rm Y}}:=\left\{\right(x,y)|y\in \mathrm{{\rm Y}}(x\left)\right\},\mathrm{dom}\mathrm{{\rm Y}}:=\left\{x|\mathrm{{\rm Y}}\right(x)\ne \mathrm{\varnothing }\}$to represent the graph and the domain of Υ, respectively. For a set $S\subset {\mathbb{R}}^m$, we define the set-valued mappings $\mathrm{{\rm Y}}+S,\mathrm{{\rm Y}}-S:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^m$ by means of $\mathrm{\forall x}\in {\mathbb{R}}^n:(\mathrm{{\rm Y}}+S)\left(x\right):=\mathrm{{\rm Y}}\left(x\right)+S,(\mathrm{{\rm Y}}-S)\left(x\right):=\mathrm{{\rm Y}}\left(x\right)-S$for brevity of notation. A set-valued mapping is referred to as polyhedral whenever its graph can be represented as the union of finitely many convex polyhedral sets.

For fixed $\overline{x}\in \mathrm{dom}\mathrm{{\rm Y}}$, we call Υ closed at $\overline{x}$ whenever for each sequence $\left(\right(x^k,y^k){)}_{k\in \mathbb{N}}\subset \mathrm{gph}\mathrm{{\rm Y}}$ and each $\overline{y}\in {\mathbb{R}}^m$ such that $x^k\to \overline{x}$ and $y^k\to \overline{y}$, we also have $\overline{y}\in \mathrm{{\rm Y}}\left(\overline{x}\right)$. Note that this property is inherent whenever $\mathrm{gph}\mathrm{{\rm Y}}$ is closed. Additionally, whenever Υ is closed at $\overline{x}$, $\mathrm{{\rm Y}}\left(\overline{x}\right)$ is a closed set. Observe that closedness of $\mathrm{gph}\mathrm{{\rm Y}}$ does not necessarily give closedness of $\mathrm{dom}\mathrm{{\rm Y}}$. However, closedness of $\mathrm{dom}\mathrm{{\rm Y}}$ and closedness of Υ at each point from $\mathrm{dom}\mathrm{{\rm Y}}$ give closedness of $\mathrm{gph}\mathrm{{\rm Y}}$. We refer to Υ as locally bounded at $\overline{x}$ if there exist a bounded set $B\subset {\mathbb{R}}^m$ and some neighborhood $U\subset {\mathbb{R}}^n$ of $\overline{x}$ such that $\mathrm{{\rm Y}}\left(x\right)\subset B$ for all $x\in U$. Furthermore, Υ is called lower semicontinuous at $\overline{x}$ (in Berge's sense) w.r.t. $\mathrm{\Omega }\subset {\mathbb{R}}^n$ whenever for each open set $V\subset {\mathbb{R}}^m$ such that $\mathrm{{\rm Y}}\left(\overline{x}\right)\cap V\ne \mathrm{\varnothing }$, there is neighborhood $U\subset {\mathbb{R}}^n$ of $\overline{x}$ such that $\mathrm{{\rm Y}}\left(x\right)\cap V\ne \mathrm{\varnothing }$ for all $x\in \mathrm{\Omega }\cap U$. For some fixed pair $(\overline{x},\overline{y})\in \mathrm{gph}\mathrm{{\rm Y}}$, we say that Υ is inner semicontinuous at $(\overline{x},\overline{y})$ w.r.t. $\mathrm{\Omega }\subset {\mathbb{R}}^n$ whenever for each neighborhood $V\subset {\mathbb{R}}^m$ of $\overline{y}$, there exists a neighborhood $U\subset {\mathbb{R}}^n$ of $\overline{x}$ such that $\mathrm{{\rm Y}}\left(x\right)\cap V\ne \mathrm{\varnothing }$ for all $x\in \mathrm{\Omega }\cap U$. If $\mathrm{\Omega }:={\mathbb{R}}^n$ can be chosen, Υ is called inner semicontinuous at $(\overline{x},\overline{y})$ for brevity. Note that Υ is lower semicontinuous at $\overline{x}$ w.r.t. Ω if and only if it is inner semicontinuous at $(\overline{x},y)$ w.r.t. Ω for each $y\in \mathrm{{\rm Y}}\left(\overline{x}\right)$. Furthermore, observe that Υ is inner semicontinuous at $(\overline{x},\overline{y})$ w.r.t. Ω if and only if for each sequence $(x^k{)}_{k\in \mathbb{N}}\subset \mathrm{\Omega }$ such that $x^k\to \overline{x}$, there exists a sequence $(y^k{)}_{k\in \mathbb{N}}\subset {\mathbb{R}}^m$ satisfying $y^k\to \overline{y}$ and $y^k\in \mathrm{{\rm Y}}\left(x^k\right)$ for all large enough $k\in \mathbb{N}$. Let us also mention that Υ is referred to as inner semicompact at $\overline{x}$ w.r.t. Ω if for each sequence $(x^k{)}_{k\in \mathbb{N}}\subset \mathrm{\Omega }$ with $x^k\to \overline{x}$, there are a subsequence $(x^{k_{\ell }}{)}_{\ell \in \mathbb{N}}$ and a convergent sequence $(y^{\ell }{)}_{\ell \in \mathbb{N}}$ such that $y^{\ell }\in \mathrm{{\rm Y}}\left(x^{k_{\ell }}\right)$ holds for all $\ell \in \mathbb{N}$. If $\mathrm{\Omega }:={\mathbb{R}}^n$ can be chosen, Υ is called inner semicompact at $\overline{x}$. Clearly, if Υ is locally bounded at $\overline{x}$, then it is inner semicompact at $\overline{x}$ w.r.t. each subset of ${\mathbb{R}}^n$. Furthermore, whenever there exists $\overline{y}\in \mathrm{{\rm Y}}\left(\overline{x}\right)$ such that Υ is inner semicontinuous at $(\overline{x},\overline{y})$ w.r.t. Ω, then Υ is inner semicompact at $\overline{x}$ w.r.t. Ω. In the definitions of lower and inner semicontinuity as well as inner semicompactness, special emphasis is laid on settings where $\mathrm{\Omega }\in \{{\mathbb{R}}^n,\mathrm{dom}\mathrm{{\rm Y}}\}$.

In the subsequently stated lemma, we characterize the closedness of the graph of set-valued mappings of special type.

Lemma 2.1

For a function $\upsilon :{\mathbb{R}}^n\to \overline{\mathbb{R}}$, we consider the set-valued mapping $\mathrm{{\rm Y}}:{\mathbb{R}}^n\rightrightarrows \mathbb{R}$ given by $\mathrm{\forall x}\in {\mathbb{R}}^n:\mathrm{{\rm Y}}\left(x\right):=\left\{\upsilon \right(x\left)\right\}-{\mathbb{R}}_{+}$with the conventions $\left\{\mathrm{\infty }\right\}-{\mathbb{R}}_{+}:=\mathbb{R}$ and $\{-\mathrm{\infty }\}-{\mathbb{R}}_{+}:=\mathrm{\varnothing }$. Then $\mathrm{gph}\mathrm{{\rm Y}}$ is closed if and only if υ is upper semicontinuous.

Proof.

We observe that $\mathrm{gph}\mathrm{{\rm Y}}$ equals $\mathrm{hypo}\upsilon :=\left\{\right(x,\alpha )|\alpha \le \upsilon (x\left)\right\}$, the so-called hypograph of υ, and $\mathrm{hypo}\upsilon$ is closed if and only if υ is an upper semicontinuous function, see e.g. [39, Theorem 1.6].

For two given set-valued mappings ${\mathrm{{\rm Y}}}_1:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^m$ and ${\mathrm{{\rm Y}}}_2:{\mathbb{R}}^m\rightrightarrows {\mathbb{R}}^p$, their composition ${\mathrm{{\rm Y}}}_2\circ {\mathrm{{\rm Y}}}_1:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^p$ is defined by $\mathrm{\forall x}\in {\mathbb{R}}^n:({\mathrm{{\rm Y}}}_2\circ {\mathrm{{\rm Y}}}_1)\left(x\right):=\bigcup _{y\in {\mathrm{{\rm Y}}}_1\left(x\right)}{\mathrm{{\rm Y}}}_2\left(y\right).$The associated so-called intermediate mapping $\mathrm{\Xi }:{\mathbb{R}}^n\times {\mathbb{R}}^p\rightrightarrows {\mathbb{R}}^m$ is given by $\mathrm{\forall x}\in {\mathbb{R}}^n\mathrm{\forall z}\in {\mathbb{R}}^p:\mathrm{\Xi }(x,z):=\{y\in {\mathrm{{\rm Y}}}_1(x)|z\in {\mathrm{{\rm Y}}}_2(y\left)\right\}$and plays an important role for the analysis of the underlying composition, see e.g. [46, Section 5.3] or [38, Section 3.1.2].

2.1.3. Variational analysis and generalized differentiation

The following definitions are taken from [38,39].

For some set $\mathrm{\Omega }\subset {\mathbb{R}}^n$, which is closed locally around some point $\overline{x}\in \mathrm{\Omega }$, we call $N_{\mathrm{\Omega }}\left(\overline{x}\right):=\left\{\eta \right|\begin{array}{rl}& \mathrm{\exists }(x^k{)}_{k\in \mathbb{N}},(y^k{)}_{k\in \mathbb{N}}\subset {\mathbb{R}}^n\mathrm{\exists }({\alpha }_k{)}_{k\in \mathbb{N}}\subset (0,\mathrm{\infty }):\\ & x^k\to \overline{x},{\alpha }_k(x^k-y^k)\to \eta ,y^k\in {\mathrm{\Pi }}_{\mathrm{\Omega }}\left(x^k\right)\mathrm{\forall k}\in \mathbb{N}\end{array}\}$the limiting (or Mordukhovich) normal cone to Ω at $\overline{x}$. Above, ${\mathrm{\Pi }}_{\mathrm{\Omega }}:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^n$ denotes the projector of Ω which is given by $\mathrm{\forall x}\in {\mathbb{R}}^n:{\mathrm{\Pi }}_{\mathrm{\Omega }}\left(x\right):={\mathrm{argmin}}_y\{\Vert y-x\Vert |y\in \mathrm{\Omega }\}.$We note that $N_{\mathrm{\Omega }}\left(\overline{x}\right)$ is a closed cone which reduces to the classical normal cone of convex analysis as soon as Ω is a convex set.

For a set-valued mapping $\mathrm{{\rm Y}}:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^m$ and some point $(\overline{x},\overline{y})\in \mathrm{gph}\mathrm{{\rm Y}}$ where $\mathrm{gph}\mathrm{{\rm Y}}$ is locally closed, the set-valued mapping $D^{\ast }\mathrm{{\rm Y}}(\overline{x},\overline{y}):{\mathbb{R}}^m\rightrightarrows {\mathbb{R}}^n$ given by $\mathrm{\forall }{y}^{\ast }\in {\mathbb{R}}^m:D^{\ast }\mathrm{{\rm Y}}(\overline{x},\overline{y})\left(y^{\ast }\right):=\left\{x^{\ast }|\right(x^{\ast },-y^{\ast })\in N_{\mathrm{gph}\mathrm{{\rm Y}}}(\overline{x},\overline{y}\left)\right\}$is referred to as the (limiting or Mordukhovich) coderivative of Υ at $(\overline{x},\overline{y})$. In the case of a single-valued continuous mapping $\upsilon :{\mathbb{R}}^n\to {\mathbb{R}}^m$, which can naturally be interpreted as a singleton-valued set-valued mapping, and $\overline{x}\in {\mathbb{R}}^n$, we make use of $D^{\ast }\upsilon \left(\overline{x}\right):=D^{\ast }\mathrm{{\rm Y}}(\overline{x},\upsilon (\overline{x}\left)\right)$ for brevity of notation. If the mapping υ is continuously differentiable at $\overline{x}$, then $D^{\ast }\upsilon \left(\overline{x}\right)\left(y^{\ast }\right)=\left\{{\upsilon }^{\prime }\right(\overline{x}{)}^{\mathrm{\top }}{y}^{\ast }\}$ is valid for all $y^{\ast }\in {\mathbb{R}}^m$.

2.2. Foundations of multiobjective optimization

For a continuous vector function $F:{\mathbb{R}}^m\to {\mathbb{R}}^p$, a nonempty, closed set $S\subset {\mathbb{R}}^m$, and some closed, convex, pointed cone $\mathcal{K}\subset {\mathbb{R}}^p$ with nonempty interior, we investigate the rather general multiobjective optimization problem (MOP) $min{}_{\mathcal{K}}\left\{F\right(y)|y\in S\}.$(MOP) Recall that some feasible point $\overline{y}\in S$ of (MOP(MOP) $min{}_{\mathcal{K}}\left\{F\right(y)|y\in S\}.$(MOP) ) is called efficient (weakly efficient) whenever $\left(\right\{F\left(\overline{y}\right)\}-\mathcal{K}\setminus \{0\left\}\right)\cap F\left(S\right)=\mathrm{\varnothing }\left(\right(\left\{F\right(\overline{y}\left)\right\}-\mathrm{int}\mathcal{K})\cap F(S)=\mathrm{\varnothing })$holds. The set of all efficient (weakly efficient) points of (MOP(MOP) $min{}_{\mathcal{K}}\left\{F\right(y)|y\in S\}.$(MOP) ) will be denoted by $\mathcal{E}(F,S,\mathcal{K})$ (${\mathcal{E}}_{\mathrm{w}}(F,S,\mathcal{K})$). Further, $\mathcal{N}(F,S,\mathcal{K}):=F\left(\mathcal{E}\right(F,S,\mathcal{K}\left)\right)$ (${\mathcal{N}}_{\mathrm{w}}(F,S,\mathcal{K}):=F\left({\mathcal{E}}_{\mathrm{w}}\right(F,S,\mathcal{K}\left)\right)$) is referred to as the nondominated (weakly nondominated) set of (MOP(MOP) $min{}_{\mathcal{K}}\left\{F\right(y)|y\in S\}.$(MOP) ). Let us mention that the computation of efficient (weakly efficient) points makes knowledge of F, S, and $\mathcal{K}$ necessary, while, in principle, nondominated (weakly nondominated) points can be characterized from knowledge of $F\left(S\right)$ and $\mathcal{K}$ only. Clearly, the inclusions $\mathcal{E}(F,S,\mathcal{K})\subset {\mathcal{E}}_{\mathrm{w}}(F,S,\mathcal{K})$ and $\mathcal{N}(F,S,\mathcal{K})\subset {\mathcal{N}}_{\mathrm{w}}(F,S,\mathcal{K})$ hold by definition. Let us also point out that continuity of F and the openness of $\mathrm{int}\mathcal{K}$ give closedness of the sets ${\mathcal{E}}_{\mathrm{w}}(F,S,\mathcal{K})$ and ${\mathcal{N}}_{\mathrm{w}}(F,S,\mathcal{K})$. In contrast, the sets $\mathcal{E}(F,S,\mathcal{K})$ and $\mathcal{N}(F,S,\mathcal{K})$ do not need to be closed, see Example 2.3 below. Let us also note that the closedness of ${\mathcal{E}}_{\mathrm{w}}(F,S,\mathcal{K})$ and ${\mathcal{N}}_{\mathrm{w}}(F,S,\mathcal{K})$ guarantees $\mathrm{cl}\mathcal{E}(F,S,\mathcal{K})\subset {\mathcal{E}}_{\mathrm{w}}(F,S,\mathcal{K})$ and $\mathrm{cl}\mathcal{N}(F,S,\mathcal{K})\subset {\mathcal{N}}_{\mathrm{w}}(F,S,\mathcal{K})$, and these inclusions can be strict. In the case where $p:=1$, there is no difference between efficiency and weak efficiency as well as nondominance and weak nondominance, respectively, since the only cones satisfying our requirements would be ${\mathbb{R}}_{+}$ and ${\mathbb{R}}_{-}$ in this case.

In this paper, we are concerned with so-called domination properties of (MOP(MOP) $min{}_{\mathcal{K}}\left\{F\right(y)|y\in S\}.$(MOP) ), see e.g. [44,45,47].

Definition 2.2

We say that (MOP(MOP) $min{}_{\mathcal{K}}\left\{F\right(y)|y\in S\}.$(MOP) ) possesses the domination property (the weak domination property) if the following inclusion is valid: $F\left(S\right)\subset \mathcal{N}(F,S,\mathcal{K})+\mathcal{K}\left(F\right(S)\subset {\mathcal{N}}_{\mathrm{w}}(F,S,\mathcal{K})+\mathcal{K}).$

It is clear from the definition that validity of the domination property also gives validity of the weak domination property. Observe that, by convexity of $\mathcal{K}$, (MOP(MOP) $min{}_{\mathcal{K}}\left\{F\right(y)|y\in S\}.$(MOP) ) possesses the domination property (the weak domination property) if and only if $F\left(S\right)+\mathcal{K}=\mathcal{N}(F,S,\mathcal{K})+\mathcal{K}\left(F\right(S)+\mathcal{K}={\mathcal{N}}_{\mathrm{w}}(F,S,\mathcal{K})+\mathcal{K}).$Thus, the discussion at the beginning of [24, Section 4] is completely superfluous. Sufficient criteria for the domination and weak domination property can be found in [44,47]. Exemplary, let us mention that (MOP(MOP) $min{}_{\mathcal{K}}\left\{F\right(y)|y\in S\}.$(MOP) ) possesses the domination property whenever F is continuous while S is compact. This follows by combining [44, Theorem 2] and [2, Theorem 3.38]. Furthermore, we also want to point the reader's attention to the fact that even in the presence of the domination property, $\mathcal{E}(F,S,\mathcal{K})$ and $\mathcal{N}(F,S,\mathcal{K})$ might be non-closed.

Example 2.3

Let $F:{\mathbb{R}}^2\to {\mathbb{R}}^2$ be the identity, let $S\subset {\mathbb{R}}^2$ be given by $S:=\{y\in {\mathbb{R}}_{+}^2|y_1^2+y_2^2\ge 1\}\cup \left(\right[-1,0]\times [1,\mathrm{\infty }\left)\right),$and let $\mathcal{K}:={\mathbb{R}}_{+}^2$. Then we find $\mathcal{E}(F,S,\mathcal{K})=\mathcal{N}(F,S,\mathcal{K})=\left\{\right(\cos t,\sin t)\in {\mathbb{R}}^2|t\in [0,\pi /2\left)\right\}\cup \left\{\right(-1,1\left)\right\}.$On the one hand, this can be used to see that the associated problem (MOP(MOP) $min{}_{\mathcal{K}}\left\{F\right(y)|y\in S\}.$(MOP) ) possesses the domination property. On the other hand, $\mathcal{E}(F,S,\mathcal{K})$ and $\mathcal{N}(F,S,\mathcal{K})$ are not closed.

It follows from [48, Example 3.4.2] that the efficient set $\mathcal{E}(F,S,\mathcal{K})$ might not be closed even if the set $F\left(S\right)$ is compact and convex. However, for instance in [49, Lemma 7.1], it is shown that if $\mathcal{K}:={\mathbb{R}}_{+}^p$ and all functions $F_1,\dots ,F_p$ are strictly quasiconvex while S is convex, then $\mathcal{E}(F,S,\mathcal{K})$ coincides with ${\mathcal{E}}_{\mathrm{w}}(F,S,\mathcal{K})$ and so $\mathcal{E}(F,S,\mathcal{K})$ is closed since ${\mathcal{E}}_{\mathrm{w}}(F,S,\mathcal{K})$ is closed. This result can also be extended to more general ordering cones.

Lemma 2.4

Let F be $\mathcal{K}$-strictly-quasiconvex, and let S be convex. Then $\mathcal{E}(F,S,\mathcal{K})={\mathcal{E}}_{\mathrm{w}}(F,S,\mathcal{K})$ and $\mathcal{N}(F,S,\mathcal{K})={\mathcal{N}}_{\mathrm{w}}(F,S,\mathcal{K})$.

Proof.

We only show equality of the efficient and weakly efficient set. This automatically gives equivalence of the nondominated and weakly nondominated set. Since $\mathcal{E}(F,S,\mathcal{K})\subset {\mathcal{E}}_{\mathrm{w}}(F,S,\mathcal{K})$ holds in general, we only need to prove the converse inclusion holds true. Fix $\overline{y}\in {\mathcal{E}}_{\mathrm{w}}(F,S,\mathcal{K})$ and suppose that $\overline{y}\notin \mathcal{E}(F,S,\mathcal{K})$. Then we find some $\tilde{y}\in S\setminus \left\{\overline{y}\right\}$ satisfying $F\left(\tilde{y}\right)\in \left\{F\right(\overline{y}\left)\right\}-\mathcal{K}\setminus \left\{0\right\}$. For all $\alpha \in (0,1)$, $\alpha \tilde{y}+(1-\alpha )\overline{y}\in S$ follows by convexity of S, and since F is $\mathcal{K}$-strictly-quasiconvex, we obtain $F(\alpha \tilde{y}+(1-\alpha \left)\overline{y}\right)\in \left(\right\{F\left(\overline{y}\right)\}-\mathrm{int}\mathcal{K})\cap F\left(S\right).$This is a contradiction to $\overline{y}\in {\mathcal{E}}_{\mathrm{w}}(F,S,\mathcal{K})$.

3. Value function models in multiobjective bilevel optimization

In order to make (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ) explicit, one has to specify the precise meaning behind the optimization goals at the (upper- and) lower-level stage. Therefore, let us define set-valued mappings $\mathrm{\Psi },{\mathrm{\Psi }}_{\mathrm{w}}:{\mathbb{R}}^{\mathrm{n}}\rightrightarrows {\mathbb{R}}^{\mathrm{m}}$ and $\mathrm{\Phi },{\mathrm{\Phi }}_{\mathrm{w}}:{\mathbb{R}}^{\mathrm{n}}\rightrightarrows {\mathbb{R}}^{\mathrm{q}}$ by means of $\begin{array}{rlrlr}\mathrm{\Psi }\left(x\right)& :=\mathcal{E}\left(f\right(x,\cdot ),\mathrm{\Gamma }(x),\mathcal{C}),& \mathrm{\Phi }\left(x\right)& :=\mathcal{N}\left(f\right(x,\cdot ),\mathrm{\Gamma }(x),\mathcal{C}),& \\ {\mathrm{\Psi }}_{\mathrm{w}}\left(\mathrm{x}\right)& :={\mathcal{E}}_{\mathrm{w}}\left(f\right(x,\cdot ),\mathrm{\Gamma }(x),\mathcal{C}),& {\mathrm{\Phi }}_{\mathrm{w}}\left(\mathrm{x}\right)& :={\mathcal{N}}_{\mathrm{w}}\left(f\right(x,\cdot ),\mathrm{\Gamma }(x),\mathcal{C})& \end{array}$for each $x\in {\mathbb{R}}^n$. Note that the identities $\mathrm{dom}\mathrm{\Psi }=\mathrm{dom}\mathrm{\Phi },\mathrm{dom}{\mathrm{\Psi }}_{\mathrm{w}}=\mathrm{dom}{\mathrm{\Phi }}_{\mathrm{w}}$hold by definition of these mappings. We will refer to Ψ and ${\mathrm{\Psi }}_{\mathrm{w}}$ as the efficiency and weak efficiency mapping associated with (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ), respectively. Similarly, Φ and ${\mathrm{\Phi }}_{\mathrm{w}}$ are called the frontier and weak frontier mapping associated with (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ). For brevity of notation, we further introduce the set-valued mapping $\mathrm{\Sigma }:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^q$ given by (3) $\mathrm{\forall x}\in {\mathbb{R}}^n:\mathrm{\Sigma }\left(x\right):=f(x,\mathrm{\Gamma }(x\left)\right).$(3) In the course of this note, we are mainly concerned with the optimization problems $\begin{array}{rl}& \underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \mathrm{\Psi }(x\left)\right\},(\mathcal{E}-\mathrm{BPP})\\ & \underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in {\mathrm{\Psi }}_{\mathrm{w}}(\mathrm{x}\left)\right\}({\mathcal{E}}_{\mathrm{w}}-\mathrm{BPP})\end{array}$as well as their associated respective value function reformulations $\begin{array}{rl}& \underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \mathrm{\Phi }(x),y\in \mathrm{\Gamma }(x\left)\right\},\left(\mathcal{E}-\mathrm{VFR}\right)\\ & \underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in {\mathrm{\Phi }}_{\mathrm{w}}(\mathrm{x}),\mathrm{y}\in \mathrm{\Gamma }(\mathrm{x}\left)\right\}({\mathcal{E}}_{\mathrm{w}}-\mathrm{VFR})\end{array}$which are apparently equivalent.

Proposition 3.1

(a)	For each $x\in {\mathbb{R}}^n$, $\mathrm{\Psi }\left(x\right)=\{y\in \mathrm{\Gamma }(x\left)|f\right(x,y)\in \mathrm{\Phi }(x\left)\right\}$ holds true.
(b)	For each $x\in {\mathbb{R}}^n$, ${\mathrm{\Psi }}_{\mathrm{w}}\left(\mathrm{x}\right)=\{\mathrm{y}\in \mathrm{\Gamma }(\mathrm{x}\left)|\mathrm{f}\right(\mathrm{x},\mathrm{y})\in {\mathrm{\Phi }}_{\mathrm{w}}(\mathrm{x}\left)\right\}$ holds true.

Proof.

Let us start with the proof of the first assertion. Indeed, if $y\in \mathrm{\Psi }\left(x\right)$ holds for some $x\in {\mathbb{R}}^n$, then $y\in \mathrm{\Gamma }\left(x\right)$ and $f(x,y)\in f(x,\mathrm{\Psi }(x\left)\right)=\mathrm{\Phi }\left(x\right)$ follow trivially, i.e. the inclusion ⊂ is obvious. For the proof of the converse inclusion $\supset$, pick $y\in \mathrm{\Gamma }\left(x\right)$ such that $f(x,y)\in \mathrm{\Phi }\left(x\right)$ for some $x\in {\mathbb{R}}^n$. By definition of Φ, we find $y^{\prime }\in \mathrm{\Psi }\left(x\right)$ such that $f(x,y)=f(x,y^{\prime })$. Since $\left(\right\{f(x,y^{\prime })\}-\mathcal{C}\setminus \{0\left\}\right)\cap \mathrm{\Sigma }\left(x\right)=\mathrm{\varnothing }$, we also have $\left(\right\{f(x,y)\}-\mathcal{C}\setminus \{0\left\}\right)\cap \mathrm{\Sigma }\left(x\right)=\mathrm{\varnothing }$, and this gives $y\in \mathrm{\Psi }\left(x\right)$.

The proof of the second assertion is completely analogous.

Corollary 3.2

(a)	Problems ($\mathcal{E}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ) and ($\mathcal{E}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ) possess the same feasible set.
(b)	Problems (${\mathcal{E}}_{\mathrm{w}}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ) and (${\mathcal{E}}_{\mathrm{w}}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ) possess the same feasible set.

Let us note that whenever $q:=1$ and $\mathcal{C}:={\mathbb{R}}_{+}$ hold, then both problems ($\mathcal{E}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ) and (${\mathcal{E}}_{\mathrm{w}}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ) are equivalent to (4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) which, up to the relation sign in the constraint involving φ, is the same as the classical value function reformulation (2(2) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\le \phi (x),y\in \mathrm{\Gamma }(x\left)\right\},$(2) ). In scalar bilevel programming, however, it has turned out to be beneficial to work with the constraint $f(x,y)\le \phi \left(x\right)$ as this opens the door to algorithmic applications and induces a sign condition for the Lagrange multiplier associated with this constraint in the context of nonsmooth optimality conditions. Noting that, for each $x\in {\mathbb{R}}^n$ and $y\in \mathrm{\Gamma }\left(x\right)$, $f(x,y)<\phi \left(x\right)$ is not possible, we can reformulate (2(2) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\le \phi (x),y\in \mathrm{\Gamma }(x\left)\right\},$(2) ) and (4(6a) $\begin{array}{rl}& \underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \mathrm{\Phi }(x)-\mathcal{C},y\in \mathrm{\Gamma }(x\left)\right\},\end{array}$(6a) ) as (5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) with the conventions $\left\{\mathrm{\infty }\right\}-{\mathbb{R}}_{+}:=\mathbb{R}$ and $\{-\mathrm{\infty }\}-{\mathbb{R}}_{+}:=\mathrm{\varnothing }$. Observing that ${\mathbb{R}}_{+}$ is the ordering cone associated with the lower-level problem in the scalar situation, this motivates to investigate (6a) $\begin{array}{rl}& \underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \mathrm{\Phi }(x)-\mathcal{C},y\in \mathrm{\Gamma }(x\left)\right\},\end{array}$(6a) (6b) $\begin{array}{rl}& \underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in {\mathrm{\Phi }}_{\mathrm{w}}(\mathrm{x})-\mathcal{C},\mathrm{y}\in \mathrm{\Gamma }(\mathrm{x}\left)\right\}\end{array}$(6b) as potential surrogates of ($\mathcal{E}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ) and (${\mathcal{E}}_{\mathrm{w}}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ), respectively. As the following results show, the addition of the lower-level ordering cone in (6a(7) $\left(\right\{f(x^k,y^k)\}-\mathrm{int}\mathcal{C})\cap \mathrm{\Sigma }\left(x^k\right)=\mathrm{\varnothing }$(7) ) and (6b(8) $\mathrm{\forall x}\in {\mathbb{R}}^n\mathrm{\forall y}\in {\mathbb{R}}^m:f(x,y):=\mathrm{Dy},\mathrm{\Gamma }\left(x\right):=\{y|\mathrm{Ax}+\mathrm{By}\le d\}.$(8) ) does not change the problem.

Proposition 3.3

(a)	For each $x\in {\mathbb{R}}^n$, $\mathrm{\Psi }\left(x\right)=\{y\in \mathrm{\Gamma }(x\left)|f\right(x,y)\in \mathrm{\Phi }(x)-\mathcal{C}\}$ holds true.
(b)	For each $x\in {\mathbb{R}}^n$, ${\mathrm{\Psi }}_{\mathrm{w}}\left(\mathrm{x}\right)=\{\mathrm{y}\in \mathrm{\Gamma }(\mathrm{x}\left)|\mathrm{f}\right(\mathrm{x},\mathrm{y})\in {\mathrm{\Phi }}_{\mathrm{w}}(\mathrm{x})-\mathcal{C}\}$ holds true.

Proof.

Let us start with the proof of the first assertion. Clearly, as $0\in \mathcal{C}$, the inclusion ⊂ is clear from Proposition 3.1. Conversely, fix $x\in {\mathbb{R}}^n$ and $y\in \mathrm{\Gamma }\left(x\right)$ such that $f(x,y)\in \mathrm{\Phi }\left(x\right)-\mathcal{C}$. Then there exist $y^{\prime }\in \mathrm{\Psi }\left(x\right)$ and $c\in \mathcal{C}$ such that $f(x,y)=f(x,y^{\prime })-c$. By definition of efficiency, this gives c = 0, i.e. $f(x,y)=f(x,y^{\prime })\in \mathrm{\Phi }\left(x\right)$, and, hence, $y\in \mathrm{\Psi }\left(x\right)$ by Proposition 3.1.

In order to prove the second assertion, we observe that the inclusion ⊂ follows again from $0\in \mathcal{C}$ and Proposition 3.1. Thus, pick an arbitrary $x\in {\mathbb{R}}^n$ and $y\in \mathrm{\Gamma }\left(x\right)$ such that $f(x,y)\in {\mathrm{\Phi }}_{\mathrm{w}}\left(\mathrm{x}\right)-\mathcal{C}$. Then there exist $y^{\prime }\in {\mathrm{\Psi }}_{\mathrm{w}}\left(\mathrm{x}\right)$ and $c\in \mathcal{C}$ such that $f(x,y)=f(x,y^{\prime })-c$. Suppose that $y\notin {\mathrm{\Psi }}_{\mathrm{w}}\left(\mathrm{x}\right)$. Then there are $\tilde{y}\in \mathrm{\Gamma }\left(x\right)$ and $\tilde{c}\in \mathrm{int}\mathcal{C}$ such that $f(x,\tilde{y})=f(x,y)-\tilde{c}$, and this gives $f(x,\tilde{y})=f(x,y^{\prime })-(c+\tilde{c})$. As we have $c+\tilde{c}\in \mathrm{int}\mathcal{C}$, $y^{\prime }\notin {\mathrm{\Psi }}_{\mathrm{w}}\left(\mathrm{x}\right)$ follows which is a contradiction. Hence, we find $y\in {\mathrm{\Psi }}_{\mathrm{w}}\left(\mathrm{x}\right)$ which yields $f(x,y)\in {\mathrm{\Phi }}_{\mathrm{w}}\left(\mathrm{x}\right)$, i.e. $y\in {\mathrm{\Psi }}_{\mathrm{w}}\left(\mathrm{x}\right)$ due to Proposition 3.1.

Corollary 3.4

(a)

Problems ($\mathcal{E}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ) and (6a(7) $\left(\right\{f(x^k,y^k)\}-\mathrm{int}\mathcal{C})\cap \mathrm{\Sigma }\left(x^k\right)=\mathrm{\varnothing }$(7) ) possess the same feasible set.

(b)

Problems (${\mathcal{E}}_{\mathrm{w}}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ) and (6b(8) $\mathrm{\forall x}\in {\mathbb{R}}^n\mathrm{\forall y}\in {\mathbb{R}}^m:f(x,y):=\mathrm{Dy},\mathrm{\Gamma }\left(x\right):=\{y|\mathrm{Ax}+\mathrm{By}\le d\}.$(8) ) possess the same feasible set.

In order to guarantee the existence of efficient or weakly efficient points for the above model problems ($\mathcal{E}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ), (${\mathcal{E}}_{\mathrm{w}}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ), ($\mathcal{E}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ), and (${\mathcal{E}}_{\mathrm{w}}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ), and to make their algorithmic treatment reasonable, it is important to guarantee that their respective feasible sets are nonempty and closed. Recalling that the mapping f is continuous, and the sets X and $\mathrm{gph}\mathrm{\Gamma }$ are assumed to be closed, this reduces to the closedness of $\mathrm{gph}\mathrm{\Psi }$, $\mathrm{gph}{\mathrm{\Psi }}_{\mathrm{w}}$, $\mathrm{gph}\mathrm{\Phi }$, and $\mathrm{gph}{\mathrm{\Phi }}_{\mathrm{w}}$, respectively. This is addressed in Sections 3.1 and 3.2 below. As we will see, $\mathrm{gph}\mathrm{\Psi }$ and $\mathrm{gph}\mathrm{\Phi }$ are not likely to be closed in several popular settings. In Section 3.3, we suggest an intermediate approach that bridges the model problems ($\mathcal{E}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ) and (${\mathcal{E}}_{\mathrm{w}}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ) (and, similarly, ($\mathcal{E}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ) and (${\mathcal{E}}_{\mathrm{w}}$-VFR)(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ) and comes along with a guaranteed closedness of the feasible set.

3.1. Closedness properties of efficiency and frontier mappings

In the following lemma, we show that ${\mathrm{\Psi }}_{\mathrm{w}}$ as well as ${\mathrm{\Phi }}_{\mathrm{w}}$ are indeed closed under mild assumptions, see e.g. [43, Corollary 3.5.6] for a related result.

Lemma 3.5

Fix $\overline{x}\in \mathrm{dom}{\mathrm{\Psi }}_{\mathrm{w}}$ and let Γ be lower semicontinuous at $\overline{x}$ w.r.t. its domain. Then ${\mathrm{\Psi }}_{\mathrm{w}}$ is closed at $\overline{x}$. If, additionally, Γ is locally bounded at $\overline{x}$, ${\mathrm{\Phi }}_{\mathrm{w}}$ is closed at $\overline{x}$.

Proof.

Let $(x^k{)}_{k\in \mathbb{N}}\subset {\mathbb{R}}^n$ and $(y^k{)}_{k\in \mathbb{N}}\subset {\mathbb{R}}^m$ as well as $\overline{y}\in {\mathbb{R}}^m$ be chosen such that $x^k\to \overline{x}$, $y^k\to \overline{y}$, and $y^k\in {\mathrm{\Psi }}_{\mathrm{w}}\left({\mathrm{x}}^{\mathrm{k}}\right)$ for all $k\in \mathbb{N}$. By definition of ${\mathrm{\Psi }}_{\mathrm{w}}$, we find $y^k\in \mathrm{\Gamma }\left(x^k\right)$ and (7) $\left(\right\{f(x^k,y^k)\}-\mathrm{int}\mathcal{C})\cap \mathrm{\Sigma }\left(x^k\right)=\mathrm{\varnothing }$(7) for all $k\in \mathbb{N}$. Since Γ is closed at $\overline{x}$, we obtain $\overline{y}\in \mathrm{\Gamma }\left(\overline{x}\right)$. Suppose that $\overline{y}\notin {\mathrm{\Psi }}_{\mathrm{w}}\left(\overline{\mathrm{x}}\right)$. Then there are $\tilde{y}\in \mathrm{\Gamma }\left(\overline{x}\right)$ and $\overline{z}\in \mathrm{int}\mathcal{C}$ such that $f(\overline{x},\overline{y})-\overline{z}=f(\overline{x},\tilde{y})$. Noting that $(x^k{)}_{k\in \mathbb{N}}\subset \mathrm{dom}{\mathrm{\Psi }}_{\mathrm{w}}\subset \mathrm{dom}\mathrm{\Gamma }$, the assumed lower semicontinuity of Γ yields the existence of a sequence $({\tilde{y}}^k{)}_{k\in \mathbb{N}}\subset {\mathbb{R}}^m$ such that ${\tilde{y}}^k\to \tilde{y}$ and ${\tilde{y}}^k\in \mathrm{\Gamma }\left(x^k\right)$ for all $k\in \mathbb{N}$. We set $z^k:=f(x^k,y^k)-f(\overline{x},\overline{y})-f(x^k,{\tilde{y}}^k)+f(\overline{x},\tilde{y})+\overline{z}$for each $k\in \mathbb{N}$. Continuity of f gives us $z^k\to \overline{z}$, and due to $\overline{z}\in \mathrm{int}\mathcal{C}$, $z^k\in \mathrm{int}\mathcal{C}$ holds for all large enough $k\in \mathbb{N}$. Some rearrangements yield $f(x^k,y^k)-z^k=f(x^k,{\tilde{y}}^k)+\left(f\right(\overline{x},\overline{y})-\overline{z}-f(\overline{x},\tilde{y}\left)\right)=f(x^k,{\tilde{y}}^k),$i.e. $\left(\right\{f(x^k,y^k)\}-\mathrm{int}\mathcal{C})\cap \mathrm{\Sigma }\left(x^k\right)\ne \mathrm{\varnothing }$for large enough $k\in \mathbb{N}$, contradicting (7(9) $\underset{y}{min}\{{\mathtt{e}}^{\mathrm{\top }}\mathrm{Dy}|A{x}^k+\mathrm{By}\le d,\mathrm{Dy}\le D{y}^k\}$(9) ). Hence, $\overline{y}\in {\mathrm{\Psi }}_{\mathrm{w}}\left(\overline{\mathrm{x}}\right)$ follows, i.e. ${\mathrm{\Psi }}_{\mathrm{w}}$ is closed at $\overline{x}$.

In order to verify the closedness of ${\mathrm{\Phi }}_{\mathrm{w}}$ at $\overline{x}$, we pick sequences $(x^k{)}_{k\in \mathbb{N}}\subset {\mathbb{R}}^n$ and $(z^k{)}_{k\in \mathbb{N}}\subset {\mathbb{R}}^p$ such that $x^k\to \overline{x}$, $z^k\in {\mathrm{\Phi }}_{\mathrm{w}}\left({\mathrm{x}}^{\mathrm{k}}\right)$ for all $k\in \mathbb{N}$, and $z^k\to \overline{z}$ for some $\overline{z}\in {\mathbb{R}}^p$. This gives the existence of $(y^k{)}_{k\in \mathbb{N}}\subset {\mathbb{R}}^m$ such that $y^k\in {\mathrm{\Psi }}_{\mathrm{w}}\left({\mathrm{x}}^{\mathrm{k}}\right)$ and $f(x^k,y^k)=z^k$ for all $k\in \mathbb{N}$. Local boundedness of Γ at $\overline{x}$ guarantees $y^k\to \overline{y}$ along a subsequence (without relabeling) for some $\overline{y}\in {\mathbb{R}}^m$. The first part of the proof ensures $\overline{y}\in {\mathrm{\Psi }}_{\mathrm{w}}\left(\overline{\mathrm{x}}\right)$, and continuity of f yields $\overline{z}=f(\overline{x},\overline{y})$, i.e. $\overline{z}\in {\mathrm{\Phi }}_{\mathrm{w}}\left(\overline{\mathrm{x}}\right)$, showing the closedness of ${\mathrm{\Phi }}_{\mathrm{w}}$ at $\overline{x}$.

As a corollary, we can state conditions which guarantee the closedness of $\mathrm{gph}{\mathrm{\Psi }}_{\mathrm{w}}$ and $\mathrm{gph}{\mathrm{\Phi }}_{\mathrm{w}}$.

Corollary 3.6

Let $\mathrm{dom}\mathrm{\Gamma }$ be closed, and let Γ be locally bounded and lower semicontinuous w.r.t. its domain at each point from $\mathrm{dom}\mathrm{\Gamma }$. Then $\mathrm{gph}{\mathrm{\Psi }}_{\mathrm{w}}$ and $\mathrm{gph}{\mathrm{\Phi }}_{\mathrm{w}}$ are closed.

Proof.

The closedness of $\mathrm{gph}\mathrm{\Gamma }$ and the assumed local boundedness guarantee that, for each $\overline{x}\in \mathrm{dom}\mathrm{\Gamma }$, $\mathrm{\Gamma }\left(\overline{x}\right)$ is nonempty and compact. Thus, [2, Theorem 6.3] shows $\mathrm{dom}{\mathrm{\Psi }}_{\mathrm{w}}=\mathrm{dom}\mathrm{\Gamma }$, i.e. $\mathrm{dom}{\mathrm{\Psi }}_{\mathrm{w}}$ and $\mathrm{dom}{\mathrm{\Phi }}_{\mathrm{w}}$ are closed. Furthermore, Lemma 3.5 shows that ${\mathrm{\Psi }}_{\mathrm{w}}$ and ${\mathrm{\Phi }}_{\mathrm{w}}$ are closed at each point of their domain, respectively. Taking these facts together, the assertion follows.

Let us relate these findings to the classical results from [50] which apply to the scalar situation $q:=1$ and $\mathcal{C}:={\mathbb{R}}_{+}$. Then the closedness of ${\mathrm{\Psi }}_{\mathrm{w}}$ corresponds to the closedness of the classical solution mapping while the closedness of ${\mathrm{\Phi }}_{\mathrm{w}}$ corresponds to continuity (w.r.t. the domain) of the so-called optimal value function. Consulting [50, Theorems 4.2.1, 4.2.2], these properties are obtained under continuity of the objective function and lower semicontinuity of the constraint mapping Γ. In this regard, Lemma 3.5 amounts to a reasonable generalization to the vector-valued situation.

Remark 3.7

Recall that the efficient set as well as the associated nondominated set in multiobjective optimization do not need to be closed even in situations where all objective functions are continuous and the feasible set is compact. This issue naturally extends to the parametric setting. Typically, the set-valued mappings Ψ and Φ do not possess closed graphs in many situations.

Let us inspect some simple examples.

Example 3.8

We consider (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ) with $n:=m:=1,q:=2$, $\mathcal{C}:={\mathbb{R}}_{+}^2$, and $\mathrm{\forall x},y\in \mathbb{R}:f(x,y):=(\sin y,y),\mathrm{\Gamma }\left(x\right):=[0,2\pi ].$The efficiency map is given by $\mathrm{\forall x}\in \mathbb{R}:\mathrm{\Psi }\left(x\right)=\left\{0\right\}\cup (\pi ,3\pi /2]$which is not closed for each $x\in \mathbb{R}$, whereas the weak efficiency map, given by $\mathrm{\forall x}\in \mathbb{R}:{\mathrm{\Psi }}_{\mathrm{w}}\left(x\right)=\left\{0\right\}\cup [\pi ,3\pi /2],$is closed at each $x\in \mathbb{R}$.

Example 3.9

We consider (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ) with $n:=1,m:=q:=2$, $\mathcal{C}:={\mathbb{R}}_{+}^2$, and $\mathrm{\forall x}\in \mathbb{R}\mathrm{\forall y}\in {\mathbb{R}}^2:f(x,y):=y,\mathrm{\Gamma }\left(x\right):=\{y|y_1+x{y}_2\ge 0\}.$For $x>0$, the efficient and weakly efficient points of (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ) coincide. In the case x = 0, there are no efficient point of (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ), but all points from $\left\{0\right\}\times \mathbb{R}$ are weakly efficient. Moreover, for $x<0$, there is neither an efficient nor a weakly efficient point. Hence, the efficiency map and frontier map are given by $\mathrm{\forall x}\in \mathbb{R}:\mathrm{\Psi }\left(x\right)=\mathrm{\Phi }\left(x\right)=\{\begin{array}{ll}\{y|y_1+x{y}_2=0\}& x>0,\\ \mathrm{\varnothing }& x\le 0,\end{array}$and these mappings do not possess closed graphs. Furthermore, the weak efficiency map and the weak frontier map are given by $\mathrm{\forall x}\in \mathbb{R}:{\mathrm{\Psi }}_{\mathrm{w}}\left(x\right)={\mathrm{\Phi }}_{\mathrm{w}}\left(x\right)=\{\begin{array}{ll}\{y|y_1+x{y}_2=0\}& x\ge 0,\\ \mathrm{\varnothing }& x<0\end{array}$and possess closed graphs, see Corollary 3.6 while noting that Γ is not locally bounded at each point $x\ge 0$.

Remark 3.10

Assume that, for each $x\in \mathrm{dom}\mathrm{\Gamma }$, $f(x,\cdot ):{\mathbb{R}}^m\to {\mathbb{R}}^q$ is $\mathcal{C}$-strictly-quasiconvex while $\mathrm{\Gamma }\left(x\right)$ is convex. Then, due to Lemma 2.4, we find $\mathrm{\Psi }\left(x\right)={\mathrm{\Psi }}_{\mathrm{w}}\left(\mathrm{x}\right)$ and $\mathrm{\Phi }\left(x\right)={\mathrm{\Phi }}_{\mathrm{w}}\left(\mathrm{x}\right)$ for all $x\in \mathrm{dom}\mathrm{\Gamma }$, i.e. there is no difference between the models ($\mathcal{E}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ) and (${\mathcal{E}}_{\mathrm{w}}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ), ($\mathcal{E}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ) and (${\mathcal{E}}_{\mathrm{w}}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ), or (6a(7) $\left(\right\{f(x^k,y^k)\}-\mathrm{int}\mathcal{C})\cap \mathrm{\Sigma }\left(x^k\right)=\mathrm{\varnothing }$(7) ) and (6b(8) $\mathrm{\forall x}\in {\mathbb{R}}^n\mathrm{\forall y}\in {\mathbb{R}}^m:f(x,y):=\mathrm{Dy},\mathrm{\Gamma }\left(x\right):=\{y|\mathrm{Ax}+\mathrm{By}\le d\}.$(8) ), respectively. Particularly, Lemma 2.4 and Corollary 3.6 can be used to infer closedness results on Ψ and Φ.

In the following lemma, we show that the efficiency and frontier mapping of a fully linear multiobjective parametric optimization problem possess closed graphs.

Lemma 3.11

For matrices $A\in {\mathbb{R}}^{r\times n}$, $B\in {\mathbb{R}}^{r\times m}$, $D\in {\mathbb{R}}^{q\times m}$, and $d\in {\mathbb{R}}^r$, let $f:{\mathbb{R}}^n\times {\mathbb{R}}^m\to {\mathbb{R}}^q$ and $\mathrm{\Gamma }:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^m$ be given by (8) $\mathrm{\forall x}\in {\mathbb{R}}^n\mathrm{\forall y}\in {\mathbb{R}}^m:f(x,y):=\mathrm{Dy},\mathrm{\Gamma }\left(x\right):=\{y|\mathrm{Ax}+\mathrm{By}\le d\}.$(8) Furthermore, let the ordering cone $\mathcal{C}:={\mathbb{R}}_{+}^q$ be fixed. Then the associated mappings Ψ and Φ possess closed graphs.

Proof.

We prove the closedness of $\mathrm{gph}\mathrm{\Psi }$ by combining a classical scalarization argument and results from scalar linear parametric optimization. Afterwards, the closedness of $\mathrm{gph}\mathrm{\Phi }$ is shown via Hofmann's lemma.

Fix a sequence $\left(\right(x^k,y^k){)}_{k\in \mathbb{N}}\subset \mathrm{gph}\mathrm{\Psi }$ and a pair $(\overline{x},\overline{y})\in {\mathbb{R}}^n\times {\mathbb{R}}^m$ such that $(x^k,y^k)\to (\overline{x},\overline{y})$. Let $k\in \mathbb{N}$ be fixed for the moment. Due to [1, Theorem 4.7], $y^k$ is an optimal solution of the scalar linear optimization problem (9) $\underset{y}{min}\{{\mathtt{e}}^{\mathrm{\top }}\mathrm{Dy}|A{x}^k+\mathrm{By}\le d,\mathrm{Dy}\le D{y}^k\}$(9) where $\mathtt{e}\in {\mathbb{R}}^q$ denotes the all-ones vector. The dual of this problem is given by (10) $\underset{u,v}{max}\{(d-A{x}^k{)}^{\mathrm{\top }}u+(D{y}^k{)}^{\mathrm{\top }}v|B^{\mathrm{\top }}u+D^{\mathrm{\top }}v=D^{\mathrm{\top }}\mathtt{e},u\le 0,v\le 0\}.$(10) As (9(11) $\underset{u,v}{max}\{(d-A\overline{x}{)}^{\mathrm{\top }}u+(D\overline{y}{)}^{\mathrm{\top }}v|(u,v)\in Q\}.$(11) ) possesses a solution, (10(12) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y\left)|\right(x,y)\in \mathrm{gph}\hat{\mathrm{\Psi }}\cap (X\times {\mathbb{R}}^m\left)\right\}.$(12) ) is solvable as well.

Let us inspect (10(12) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y\left)|\right(x,y)\in \mathrm{gph}\hat{\mathrm{\Psi }}\cap (X\times {\mathbb{R}}^m\left)\right\}.$(12) ) which can be interpreted as a linear parametric optimization problem with multiple parameters, namely $x^k$ and $y^k$, which merely appear in the objective function. Based on [51, Sections 7, 8], the set of parameters where this parametric problem possesses a solution is a convex polyhedron and can be partitioned into finitely many so-called regions of stability, being convex polyhedra as well, such that, in the interior of any such region, precisely one of the vertices associated with the convex polyhedron $Q:=\{(u,v)|B^{\mathrm{\top }}u+D^{\mathrm{\top }}v=D^{\mathrm{\top }}\mathtt{e},u\le 0,v\le 0\}$is optimal, while on the boundary, this vertex is still optimal but may not be the unique maximizer. Anyhow, since there exist only finitely many regions of stability, there is an infinite index set $N\subset \mathbb{N}$ such that all points from the subsequence $\left(\right(x^k,y^k){)}_{k\in N}$ belong to the same region of stability which is associated with some fixed vertex $(\overline{u},\overline{v})\in {\mathbb{R}}^n\times {\mathbb{R}}^m$ of Q. As all regions of stability are closed, $(\overline{u},\overline{v})$ does not only solve (10(12) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y\left)|\right(x,y)\in \mathrm{gph}\hat{\mathrm{\Psi }}\cap (X\times {\mathbb{R}}^m\left)\right\}.$(12) ) for each $k\in N$ but also the problem (11) $\underset{u,v}{max}\{(d-A\overline{x}{)}^{\mathrm{\top }}u+(D\overline{y}{)}^{\mathrm{\top }}v|(u,v)\in Q\}.$(11) Strong duality of linear programming yields ${\mathtt{e}}^{\mathrm{\top }}D{y}^k=(d-A{x}^k{)}^{\mathrm{\top }}\overline{u}+(D{y}^k{)}^{\mathrm{\top }}\overline{v}$ for each $k\in N$, and taking the limit $k{\to }_N\mathrm{\infty }$ gives ${\mathtt{e}}^{\mathrm{\top }}D\overline{y}=(d-A\overline{x}{)}^{\mathrm{\top }}\overline{u}+(D\overline{y}{)}^{\mathrm{\top }}\overline{v}$. As $\overline{y}$ is feasible to $\underset{y}{min}\{{\mathtt{e}}^{\mathrm{\top }}\mathrm{Dy}|A\overline{x}+\mathrm{By}\le d,\mathrm{Dy}\le D\overline{y}\},$which is the dual of (11), it already must be a minimizer. Applying [1, Theorem 4.7] once more gives $\overline{y}\in \mathrm{\Psi }\left(\overline{x}\right)$. This shows the closedness of $\mathrm{gph}\mathrm{\Psi }$.

In the remainder of the proof, we show that $\mathrm{gph}\mathrm{\Phi }$ is closed. Thus, let $(\overline{x},\overline{z})\in {\mathbb{R}}^n\times {\mathbb{R}}^q$ and $\left(\right(x^k,z^k){)}_{k\in \mathbb{N}}\subset \mathrm{gph}\mathrm{\Phi }$ be chosen such that $(x^k,z^k)\to (\overline{x},\overline{z})$. By definition of Φ, we find a sequence $(y^k{)}_{k\in \mathbb{N}}\subset {\mathbb{R}}^m$ such that $y^k\in \mathrm{\Psi }\left(x^k\right)$ and $z^k=D{y}^k$ hold for each $k\in \mathbb{N}$. Applying Hofmann's lemma, see e.g. [52, Lemma 3C.4], for each $k\in \mathbb{N}$ to the linear system $\mathrm{Dy}\le z^k,-\mathrm{Dy}\le -z^k,\mathrm{By}\le d-A{x}^k$

yields some ${\tilde{y}}^k\in \mathrm{\Gamma }\left(x^k\right)$ which satisfies $D{\tilde{y}}^k=z^k$ and the upper bound $\Vert {\tilde{y}}^k\Vert \le \kappa (\Vert z^k\Vert +\Vert d-A{x}^k\Vert )$ where the constant $\kappa >0$ does not depend on k. Particularly, for each $k\in \mathbb{N}$, $D{\tilde{y}}^k=z^k=D{y}^k$, i.e. ${\tilde{y}}^k\in \mathrm{\Psi }\left(x^k\right)$. Further, by boundedness of $(x^k{)}_{k\in \mathbb{N}}$ and $(z^k{)}_{k\in \mathbb{N}}$, $({\tilde{y}}^k{)}_{k\in \mathbb{N}}$ is bounded as well so we may assume ${\tilde{y}}^k\to \tilde{y}$ for some $\tilde{y}\in {\mathbb{R}}^m$. The first part of the proof gives $\tilde{y}\in \mathrm{\Psi }\left(\overline{x}\right)$, so that $D\tilde{y}=\overline{z}$ yields $\overline{z}\in \mathrm{\Phi }\left(\overline{x}\right)$. Hence, $\mathrm{gph}\mathrm{\Phi }$ is closed.

We note that the first part of the above proof has been inspired by classical arguments of Isermann who showed that all efficient points of a linear multiobjective optimization problem can be found with the aid of the weighted-sum scalarization technique where all weights are positive, see [53].

In the context of Proposition 3.3 and Corollary 3.4, the following result should be taken into account.

Lemma 3.12

Let Γ be locally bounded at each point of its domain. Then the following assertions hold.

(a)	The set $\mathrm{gph}\mathrm{\Phi }$ is closed if and only if $\mathrm{gph}(\mathrm{\Phi }-\mathcal{C})$ is closed.
(b)	The set $\mathrm{gph}{\mathrm{\Phi }}_{\mathrm{w}}$ is closed if and only if $\mathrm{gph}({\mathrm{\Phi }}_{\mathrm{w}}-\mathcal{C})$ is closed.

Proof.

We only prove the first assertion. The second one can be shown in analogous fashion while incorporating some arguments from the proof of Proposition 3.3.

We show both implications separately.

$[⟹]$ Let $\mathrm{gph}\mathrm{\Phi }$ be closed, and choose $\left(\right(x^k,z^k){)}_{k\in \mathbb{N}}\subset \mathrm{gph}(\mathrm{\Phi }-\mathcal{C})$ such that $(x^k,z^k)\to (\overline{x},\overline{z})$ for some $(\overline{x},\overline{z})\in {\mathbb{R}}^n\times {\mathbb{R}}^q$. By definition, we find sequences $(c^k{)}_{k\in \mathbb{N}}\subset \mathcal{C}$ and $(y^k{)}_{k\in \mathbb{N}}\subset {\mathbb{R}}^m$ such that $z^k=f(x^k,y^k)-c^k$ and $y^k\in \mathrm{\Psi }\left(x^k\right)$, i.e. $(x^k,f(x^k,y^k\left)\right)\in \mathrm{gph}\mathrm{\Phi }$, for each $k\in \mathbb{N}$. As the sequence $\left(\right(x^k,y^k){)}_{k\in \mathbb{N}}\subset \mathrm{gph}\mathrm{\Gamma }$ is bounded by assumption, it possesses an accumulation point $(\overline{x},\overline{y})$ which, by continuity of f and closedness of $\mathrm{gph}\mathrm{\Phi }$, satisfies $f(\overline{x},\overline{y})\in \mathrm{\Phi }\left(\overline{x}\right)$. Furthermore, $(c^k{)}_{k\in \mathbb{N}}$ possesses the accumulation point $f(\overline{x},\overline{y})-\overline{z}$ which, by closedness of $\mathcal{C}$, belongs to $\mathcal{C}$. Hence, we have shown $\overline{z}\in (\mathrm{\Phi }-\mathcal{C})\left(\overline{x}\right)$, and closedness of $\mathrm{gph}(\mathrm{\Phi }-\mathcal{C})$ follows.

$[⟸]$ Let $\mathrm{gph}(\mathrm{\Phi }-\mathcal{C})$ be closed, and choose a sequence $\left(\right(x^k,z^k){)}_{k\in \mathbb{N}}\subset \mathrm{gph}\mathrm{\Phi }$ such that $(x^k,z^k)\to (\overline{x},\overline{z})$ for some $(\overline{x},\overline{z})\in {\mathbb{R}}^n\times {\mathbb{R}}^q$. On the one hand, since $0\in \mathcal{C}$, we also have $\left(\right(x^k,z^k){)}_{k\in \mathbb{N}}\subset \mathrm{gph}(\mathrm{\Phi }-\mathcal{C})$, and $(\overline{x},\overline{z})\in \mathrm{gph}(\mathrm{\Phi }-\mathcal{C})$ follows, i.e. there are $\overline{y}\in \mathrm{\Psi }\left(\overline{x}\right)$ and $c\in \mathcal{C}$ such that $\overline{z}=f(\overline{x},\overline{y})-c$. On the other hand, there is a sequence $(y^k{)}_{k\in \mathbb{N}}\subset {\mathbb{R}}^m$ such that $z^k=f(x^k,y^k)$ and $y^k\in \mathrm{\Psi }\left(x^k\right)$ for all $k\in \mathbb{N}$. As the sequence $\left(\right(x^k,y^k){)}_{k\in \mathbb{N}}\subset \mathrm{gph}\mathrm{\Gamma }$ is bounded by assumption, it possesses an accumulation point $(\overline{x},\tilde{y})$ which, by closedness of $\mathrm{gph}\mathrm{\Gamma }$, belongs to $\mathrm{gph}\mathrm{\Gamma }$. Combining these observations, we have $f(\overline{x},\tilde{y})=\overline{z}=f(\overline{x},\overline{y})-c$. Now, $\overline{y}\in \mathrm{\Psi }\left(\overline{x}\right)$ guarantees c = 0, i.e. $\overline{z}=f(\overline{x},\overline{y})\in \mathrm{\Phi }\left(\overline{x}\right)$, and closedness of $\mathrm{gph}\mathrm{\Phi }$ follows.

3.2. Existence results in multiobjective bilevel optimization

The following example shows that the closedness of $\mathrm{gph}\hat{\mathrm{\Psi }}$ is, indeed, essential for the existence of efficient and weakly efficient points of (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ).

Example 3.13

We consider (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ) with $n:=m:=1,p:=2$, $\mathcal{K}:={\mathbb{R}}_{+}^2$, $X:=\mathbb{R}$, and $\mathrm{\forall x},y\in \mathbb{R}:F(x,y):=(\cos (y)+1,(y-\pi {)}^2).$Furthermore, we assume that $\hat{\mathrm{\Psi }}\in \{\mathrm{\Psi },{\mathrm{\Psi }}_{\mathrm{w}}\}$ is the efficiency or weak efficiency map of the parametric optimization problem from Example 3.8, i.e. $\mathrm{\forall x}\in \mathbb{R}:\mathrm{\Psi }\left(x\right)=\left\{0\right\}\cup (\pi ,3\pi /2],{\mathrm{\Psi }}_{\mathrm{w}}\left(\mathrm{x}\right)=\left\{0\right\}\cup [\pi ,3\pi /2].$In the case $\hat{\mathrm{\Psi }}:=\mathrm{\Psi }$, there is neither an efficient nor a weakly effficient point of (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ). However, in the case $\hat{\mathrm{\Psi }}:={\mathrm{\Psi }}_{\mathrm{w}}$, $\mathbb{R}\times \left\{\pi \right\}$ is a feasible subset of (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ), and these are precisely the efficient and, simultaneously, weakly efficient points of the problem.

The following result concerns the existence of efficient and weakly efficient points of (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ).

Theorem 3.14

Let $\mathrm{gph}\hat{\mathrm{\Psi }}$ be closed and $\mathrm{gph}\hat{\mathrm{\Psi }}\cap (X\times {\mathbb{R}}^m)$ be nonempty and bounded. Then there exist an efficient and a weakly efficient point of (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ).

Proof.

Problem (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ) is equivalent to (12) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y\left)|\right(x,y)\in \mathrm{gph}\hat{\mathrm{\Psi }}\cap (X\times {\mathbb{R}}^m\left)\right\}.$(12) Due to the postulated assumptions, the set $\mathrm{gph}\hat{\mathrm{\Psi }}\cap (X\times {\mathbb{R}}^m)$ is nonempty, bounded, and closed and, thus, compact. Then $F(\mathrm{gph}\hat{\mathrm{\Psi }}\cap (X\times {\mathbb{R}}^m\left)\right)$ is compact because F is continuous. The existence of an efficient point of (12(13) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \overline{\mathrm{\Phi }}(x)-\mathcal{C},y\in \mathrm{\Gamma }(x\left)\right\}$(13) ), which then is efficient and, thus, weakly efficient for (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ), follows by [2, Theorem 6.3].

Let us note that the boundedness assumption on $\mathrm{gph}\hat{\mathrm{\Psi }}\cap (X\times {\mathbb{R}}^m)$ in Theorem 3.14 is inherently satisfied whenever X and $\bigcup _{x\in X}\mathrm{\Gamma }\left(x\right)$ are bounded.

Based on Corollary 3.6, we obtain from Theorem 3.14 an existence result for efficient and weakly efficient points of (${\mathcal{E}}_{\mathrm{w}}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ).

Corollary 3.15

Let $\mathrm{dom}\mathrm{\Gamma }$ be closed, and let Γ be locally bounded and lower semicontinuous w.r.t. its domain at each point from $\mathrm{dom}\mathrm{\Gamma }$. Let $\mathrm{gph}{\mathrm{\Psi }}_{\mathrm{w}}\cap (\mathrm{X}\times {\mathbb{R}}^{\mathrm{m}})$ be nonempty and bounded. Then there exist an efficient and a weakly efficient point of (${\mathcal{E}}_{\mathrm{w}}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ).

We note that Corollary 3.15 generalizes [17, Theorem 5.5] to multiobjective bilevel optimization problems, but does not cover the situation were the lower-level variable comes from an infinite-dimensional space.

Similar as in Corollary 3.15, Theorem 3.14 gives an existence result for efficient and weakly efficient points of ($\mathcal{E}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ) and (${\mathcal{E}}_{\mathrm{w}}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ) whenever the lower-level data is fully linear as described in (8(10) $\underset{u,v}{max}\{(d-A{x}^k{)}^{\mathrm{\top }}u+(D{y}^k{)}^{\mathrm{\top }}v|B^{\mathrm{\top }}u+D^{\mathrm{\top }}v=D^{\mathrm{\top }}\mathtt{e},u\le 0,v\le 0\}.$(10) ).

Corollary 3.16

Let the lower-level data be given as in (8(10) $\underset{u,v}{max}\{(d-A{x}^k{)}^{\mathrm{\top }}u+(D{y}^k{)}^{\mathrm{\top }}v|B^{\mathrm{\top }}u+D^{\mathrm{\top }}v=D^{\mathrm{\top }}\mathtt{e},u\le 0,v\le 0\}.$(10) ) and fix $\mathcal{C}:={\mathbb{R}}_{+}^q$. Then the following assertions hold.

(a)	Let $\mathrm{gph}\mathrm{\Psi }\cap (X\times {\mathbb{R}}^m)$ be nonempty and bounded. Then there exist an efficient and a weakly efficient point of ($\mathcal{E}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ).
(b)	Let $\mathrm{gph}{\mathrm{\Psi }}_{\mathrm{w}}\cap (\mathrm{X}\times {\mathbb{R}}^{\mathrm{m}})$ be nonempty and bounded. Then there exist an efficient and a weakly efficient point of (${\mathcal{E}}_{\mathrm{w}}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ).

Proof.

The first assertion directly follows from Lemma 3.11 and Theorem 3.14. The second statement can be obtained from Corollary 3.6 and Theorem 3.14 as $\mathrm{gph}\mathrm{\Gamma }$ is a convex polyhedral set and, thus, closed, and these properties also ensure that Γ is lower semicontinuous w.r.t. its domain at each point from $\mathrm{dom}\mathrm{\Gamma }$, see e.g. [39, Example 9.35].

Remark 3.17

Taking a look back at the optimal value function reformulation for standard scalar bilevel optimization problems, we know from Lemma 2.1 that the feasible set of (5(6b) $\begin{array}{rl}& \underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in {\mathrm{\Phi }}_{\mathrm{w}}(\mathrm{x})-\mathcal{C},\mathrm{y}\in \mathrm{\Gamma }(\mathrm{x}\left)\right\}\end{array}$(6b) ) is closed if and only if φ is upper semicontinuous, and this can also be seen from (2(2) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\le \phi (x),y\in \mathrm{\Gamma }(x\left)\right\},$(2) ) as sublevel sets of extended-real-valued functions are closed if and only if they are lower semicontinuous. The latter property is clearly milder than the continuity of φ on its domain $\left\{x||\phi \right(x)|<\mathrm{\infty }\}$ which seems to be necessary for this desirable closedness when focusing on the model problem (4(6a) $\begin{array}{rl}& \underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \mathrm{\Phi }(x)-\mathcal{C},y\in \mathrm{\Gamma }(x\left)\right\},\end{array}$(6a) ).

Thus, having problems (6a(7) $\left(\right\{f(x^k,y^k)\}-\mathrm{int}\mathcal{C})\cap \mathrm{\Sigma }\left(x^k\right)=\mathrm{\varnothing }$(7) ) and (6b(8) $\mathrm{\forall x}\in {\mathbb{R}}^n\mathrm{\forall y}\in {\mathbb{R}}^m:f(x,y):=\mathrm{Dy},\mathrm{\Gamma }\left(x\right):=\{y|\mathrm{Ax}+\mathrm{By}\le d\}.$(8) ) as well as Corollary 3.4 at hand, it is seemingly reasonable to work out conditions which ensure that the graphs of $\mathrm{\Phi }-\mathcal{C}$ and ${\mathrm{\Phi }}_{\mathrm{w}}-\mathcal{C}$ are closed. This is enough to obtain the closedness of the feasible sets of (6a(7) $\left(\right\{f(x^k,y^k)\}-\mathrm{int}\mathcal{C})\cap \mathrm{\Sigma }\left(x^k\right)=\mathrm{\varnothing }$(7) ) and (6b(8) $\mathrm{\forall x}\in {\mathbb{R}}^n\mathrm{\forall y}\in {\mathbb{R}}^m:f(x,y):=\mathrm{Dy},\mathrm{\Gamma }\left(x\right):=\{y|\mathrm{Ax}+\mathrm{By}\le d\}.$(8) ) which, in turn, provides the basis for further existence results in multiobjective bilevel optimization. However, note that due to Lemma 3.12, it turns out that whenever the feasibility mapping Γ is locally bounded, then this apparently milder approach yields no new insights. Observe that this is in accordance with the aforementioned scalar situation as the optimal value function φ is lower semicontinuous on its domain whenever Γ is locally bounded.

3.3. An intermediate approach to multiobjective bilevel optimization

On the one hand, the feasible set of ($\mathcal{E}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ) is likely to be much smaller than the one of (${\mathcal{E}}_{\mathrm{w}}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ) since the solution criterion behind Ψ is sharper than the one behind ${\mathrm{\Psi }}_{\mathrm{w}}$. On the other hand, from the viewpoint of existence theory, necessary optimality conditions, and solution algorithms, a closed feasible set is desirable. The latter is given for (${\mathcal{E}}_{\mathrm{w}}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ) under mild assumptions, see Corollary 3.6, while the feasible set of ($\mathcal{E}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ) is not likely to be closed apart from selected situations.

In this subsection, we suggest yet another possibility for the choice of $\hat{\mathrm{\Psi }}$ in the general multiobjective bilevel optimization problem (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ) which induces an intermediate model between ($\mathcal{E}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ) and (${\mathcal{E}}_{\mathrm{w}}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ). To start, however, we would like to suggest an intermediate value function model which bridges ($\mathcal{E}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ) and (${\mathcal{E}}_{\mathrm{w}}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ) to some extent. There are two reasons for this approach. First, it allows for a simple definition of an intermediate solution map such that the associated models (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ) and (VFR(VFR) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \hat{\mathrm{\Phi }}(x),y\in \mathrm{\Gamma }(x\left)\right\}$(VFR) ) are equivalent. Second, noting that, in order to determine efficient and weakly efficient points of some multiobjective optimization problem, one has to compute its nondominated and weakly nondominated points first before identifying their pre-images in the feasible set, this procedure seems the more natural one.

Therefore, let us introduce $\overline{\mathrm{\Phi }}:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^q$ by means of $\mathrm{gph}\overline{\mathrm{\Phi }}:=\mathrm{cl}\mathrm{gph}\mathrm{\Phi }$and consider $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \overline{\mathrm{\Phi }}(x),y\in \mathrm{\Gamma }(x\left)\right\}.(\overline{\mathcal{E}}-\mathrm{VFR})$As $\mathrm{gph}{\mathrm{\Phi }}_{\mathrm{w}}$ is closed under mild assumptions, see e.g. Corollary 3.6, the inclusions $\mathrm{gph}\mathrm{\Phi }\subset \mathrm{gph}\overline{\mathrm{\Phi }}\subset \mathrm{gph}{\mathrm{\Phi }}_{\mathrm{w}}$ are likely to hold and would give $\mathrm{\Phi }\left(x\right)\subset \overline{\mathrm{\Phi }}\left(x\right)\subset {\mathrm{\Phi }}_{\mathrm{w}}\left(\mathrm{x}\right)$ for each $x\in {\mathbb{R}}^n$. Hence, ($\overline{\mathcal{E}}$-VFR(14) $D^{\ast }\mathrm{\Phi }(\overline{x},\overline{z})\left(z^{\ast }\right)\subset D^{\ast }(\mathrm{\Sigma }+\mathcal{C})(\overline{x},\overline{z})\left(z^{\ast }\right)$(14) ) can be interpreted as an intermediate problem between ($\mathcal{E}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ) and (${\mathcal{E}}_{\mathrm{w}}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ) since the feasible set of ($\overline{\mathcal{E}}$-VFR(14) $D^{\ast }\mathrm{\Phi }(\overline{x},\overline{z})\left(z^{\ast }\right)\subset D^{\ast }(\mathrm{\Sigma }+\mathcal{C})(\overline{x},\overline{z})\left(z^{\ast }\right)$(14) (5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ) is not smaller than the one of ($\mathcal{E}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ) and likely to be not larger than the one of (${\mathcal{E}}_{\mathrm{w}}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ). Furthermore, ($\overline{\mathcal{E}}$-VFR(14) $D^{\ast }\mathrm{\Phi }(\overline{x},\overline{z})\left(z^{\ast }\right)\subset D^{\ast }(\mathrm{\Sigma }+\mathcal{C})(\overline{x},\overline{z})\left(z^{\ast }\right)$(14) ) possesses a closed feasible set by construction of $\overline{\mathrm{\Phi }}$, and, thus, efficient and weakly efficient points of ($\overline{\mathcal{E}}$-VFR(14) $D^{\ast }\mathrm{\Phi }(\overline{x},\overline{z})\left(z^{\ast }\right)\subset D^{\ast }(\mathrm{\Sigma }+\mathcal{C})(\overline{x},\overline{z})\left(z^{\ast }\right)$(14) ) exist under standard assumptions.

Based on $\overline{\mathrm{\Phi }}$, let us introduce $\overline{\mathrm{\Psi }}:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^m$ by means of $\mathrm{\forall x}\in {\mathbb{R}}^n:\overline{\mathrm{\Psi }}\left(x\right):=\{y\in \mathrm{\Gamma }(x\left)|f\right(x,y)\in \overline{\mathrm{\Phi }}(x\left)\right\}.$In the subsequent lemma, we comment on the properties of $\overline{\mathrm{\Psi }}$. The simple proof follows by definition of $\overline{\mathrm{\Psi }}$ as well as our above discussion on the properties of $\overline{\mathrm{\Phi }}$ and is, thus, omitted.

Lemma 3.18

(a)	The set $\mathrm{gph}\overline{\mathrm{\Psi }}$ is closed.
(b)	Whenever $\mathrm{gph}{\mathrm{\Phi }}_{\mathrm{w}}$ is closed, then the inclusions $\mathrm{gph}\mathrm{\Psi }\subset \mathrm{gph}\overline{\mathrm{\Psi }}\subset \mathrm{gph}{\mathrm{\Psi }}_{\mathrm{w}}$ hold. Particularly, $\mathrm{\Psi }\left(x\right)\subset \overline{\mathrm{\Psi }}\left(x\right)\subset {\mathrm{\Psi }}_{\mathrm{w}}\left(\mathrm{x}\right)$ is valid for each $x\in {\mathbb{R}}^n$ in this situation.
(c)	We have $\mathrm{cl}\mathrm{gph}\mathrm{\Psi }\subset \mathrm{gph}\overline{\mathrm{\Psi }}$.

By definition of $\overline{\mathrm{\Psi }}$, the problem

is equivalent to ($\overline{\mathcal{E}}$-VFR(14) $D^{\ast }\mathrm{\Phi }(\overline{x},\overline{z})\left(z^{\ast }\right)\subset D^{\ast }(\mathrm{\Sigma }+\mathcal{C})(\overline{x},\overline{z})\left(z^{\ast }\right)$(14) ), and due to Lemma 3.18(b), can be seen as an intermediate model between ($\mathcal{E}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ) and (${\mathcal{E}}_{\mathrm{w}}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ). Clearly, ($\overline{\mathcal{E}}$-BPP(15) $D^{\ast }\mathrm{\Phi }(\overline{x},\overline{z})\left(z^{\ast }\right)\subset D^{\ast }\mathrm{\Sigma }(\overline{x},\overline{z})\left(z^{\ast }\right)$(15) ) has a closed feasible set and, due to Theorem 3.14, is likely to possess efficient and weakly efficient points which can be interpreted as approximately efficient and approximately weakly efficient for ($\mathcal{E}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ).

Let us state yet another motivation behind the consideration of the models ($\overline{\mathcal{E}}$-BPP(15) $D^{\ast }\mathrm{\Phi }(\overline{x},\overline{z})\left(z^{\ast }\right)\subset D^{\ast }\mathrm{\Sigma }(\overline{x},\overline{z})\left(z^{\ast }\right)$(15) ) and ($\overline{\mathcal{E}}$-VFR(14) $D^{\ast }\mathrm{\Phi }(\overline{x},\overline{z})\left(z^{\ast }\right)\subset D^{\ast }(\mathrm{\Sigma }+\mathcal{C})(\overline{x},\overline{z})\left(z^{\ast }\right)$(14) ). When solving ($\mathcal{E}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ) and ($\mathcal{E}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ) numerically with a suitable solution algorithm, respectively, one typically obtains a sequence of iterates from the solution method. Taking into account the potential non-closedness of $\mathrm{gph}\mathrm{\Psi }$ and $\mathrm{gph}\mathrm{\Phi }$, accumulation points of this sequence are not necessarily feasible to ($\mathcal{E}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ) and ($\mathcal{E}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ), respectively. However, they are likely to be feasible to ($\overline{\mathcal{E}}$-BPP(15) $D^{\ast }\mathrm{\Phi }(\overline{x},\overline{z})\left(z^{\ast }\right)\subset D^{\ast }\mathrm{\Sigma }(\overline{x},\overline{z})\left(z^{\ast }\right)$(15) ) and ($\overline{\mathcal{E}}$-VFR(14) $D^{\ast }\mathrm{\Phi }(\overline{x},\overline{z})\left(z^{\ast }\right)\subset D^{\ast }(\mathrm{\Sigma }+\mathcal{C})(\overline{x},\overline{z})\left(z^{\ast }\right)$(14) (5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ) as $\mathrm{gph}\overline{\mathrm{\Psi }}$ and $\mathrm{gph}\overline{\mathrm{\Phi }}$ are closed supersets of $\mathrm{gph}\mathrm{\Psi }$ and $\mathrm{gph}\mathrm{\Phi }$, respectively. Hence, qualitative properties of solution methods associated with ($\mathcal{E}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ) and ($\mathcal{E}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ) are naturally related to the model problems ($\overline{\mathcal{E}}$-BPP(15) $D^{\ast }\mathrm{\Phi }(\overline{x},\overline{z})\left(z^{\ast }\right)\subset D^{\ast }\mathrm{\Sigma }(\overline{x},\overline{z})\left(z^{\ast }\right)$(15) ) and ($\overline{\mathcal{E}}$-VFR(14) $D^{\ast }\mathrm{\Phi }(\overline{x},\overline{z})\left(z^{\ast }\right)\subset D^{\ast }(\mathrm{\Sigma }+\mathcal{C})(\overline{x},\overline{z})\left(z^{\ast }\right)$(14) ), respectively.

Let us point out that one does not obtain equality in Lemma 3.18(c) in general situations. Furthermore, when defining $\tilde{\mathrm{\Psi }}:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^m$ by $\mathrm{gph}\tilde{\mathrm{\Psi }}:=\mathrm{cl}\mathrm{gph}\mathrm{\Psi }$ and $\tilde{\mathrm{\Phi }}:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^q$ by $\tilde{\mathrm{\Phi }}\left(x\right):=f(x,\tilde{\mathrm{\Psi }}(x\left)\right)$ for all $x\in {\mathbb{R}}^n$, then the corresponding problems (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ) and (VFR(VFR) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \hat{\mathrm{\Phi }}(x),y\in \mathrm{\Gamma }(x\left)\right\}$(VFR) ) for $\hat{\mathrm{\Psi }}:=\tilde{\mathrm{\Psi }}$ and $\hat{\mathrm{\Phi }}:=\tilde{\mathrm{\Phi }}$, respectively, are not necessarily equivalent. We omitted this undesirable behavior by starting with the definition of a suitable intermediate frontier mapping before introducing an intermediate efficiency mapping.

Example 3.19

We consider (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ) with $n:=1,m:=q:=2$, $\mathcal{C}:={\mathbb{R}}_{+}^2$, and $\begin{array}{rl}\mathrm{\forall x}\in \mathbb{R}\mathrm{\forall y}\in {\mathbb{R}}^2:f(x,y)& :=(\sin y_1,y_2),\\ \mathrm{\Gamma }\left(x\right)& :=\mathrm{conv}\left\{\right(0,0),(2\pi ,2\pi \left)\right\}\cup \left\{\right(0,\pi \left)\right\}\end{array}$We easily find $\begin{array}{rl}\mathrm{\forall x}\in \mathbb{R}:\mathrm{\Psi }\left(x\right)& =\{y|y_1=y_2,y_2\in \{0\}\cup (\pi ,3\pi /2\left]\right\},\\ \mathrm{\Phi }\left(x\right)& =\left\{\right(\sin z,z)|z\in \{0\}\cup (\pi ,3\pi /2\left]\right\}\end{array}$which gives $\begin{array}{rl}\mathrm{\forall x}\in \mathbb{R}:\tilde{\mathrm{\Psi }}\left(x\right)& =\{y|y_1=y_2,y_2\in \{0\}\cup [\pi ,3\pi /2\left]\right\},\\ \tilde{\mathrm{\Phi }}\left(x\right)& =\left\{\right(\sin z,z)|z\in \{0\}\cup [\pi ,3\pi /2\left]\right\}.\end{array}$Observe, however, that $\mathrm{\forall x}\in \mathbb{R}:\{y\in \mathrm{\Gamma }(x\left)|f\right(x,y)\in \tilde{\mathrm{\Phi }}(x\left)\right\}=\tilde{\mathrm{\Psi }}\left(x\right)\cup \left\{\right(0,\pi \left)\right\},$i.e. the feasible set of any superordinate model (BPP(BPP) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,y\in \hat{\mathrm{\Psi }}(x\left)\right\}$(BPP) ) for $\hat{\mathrm{\Psi }}:=\tilde{\mathrm{\Psi }}$ in likely to be smaller than the feasible set of the associated value function reformulation (VFR(VFR) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \hat{\mathrm{\Phi }}(x),y\in \mathrm{\Gamma }(x\left)\right\}$(VFR) ) for $\hat{\mathrm{\Phi }}:=\tilde{\mathrm{\Phi }}$.

Let us also mention that $\begin{array}{rl}\mathrm{\forall x}\in \mathbb{R}:\overline{\mathrm{\Phi }}\left(x\right)& =\left\{\right(\sin z,z)|z\in \{0\}\cup [\pi ,3\pi /2\left]\right\}=\tilde{\mathrm{\Phi }}\left(x\right),\\ \overline{\mathrm{\Psi }}\left(x\right)& =\tilde{\mathrm{\Psi }}\left(x\right)\cup \left\{\right(0,\pi \left)\right\}\end{array}$holds in this example. This also shows $\mathrm{cl}\mathrm{gph}\mathrm{\Psi }⊊\mathrm{gph}\overline{\mathrm{\Psi }}$.

The following example shows that ($\mathcal{E}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ), (${\mathcal{E}}_{\mathrm{w}}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ), and ($\overline{\mathcal{E}}$-BPP(15) $D^{\ast }\mathrm{\Phi }(\overline{x},\overline{z})\left(z^{\ast }\right)\subset D^{\ast }\mathrm{\Sigma }(\overline{x},\overline{z})\left(z^{\ast }\right)$(15) ) may possess pairwise different solutions.

Example 3.20

Motivated by Example 2.3, for $n:=1,m:=q:=2$, and $\mathcal{C}:={\mathbb{R}}_{+}^2$, we consider the parametric multiobjective optimization problem (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ) with the data $\begin{array}{rl}\mathrm{\forall x}\in \mathbb{R}\mathrm{\forall y}\in {\mathbb{R}}^2:f(x,y)& :=y,\\ \mathrm{\Gamma }\left(x\right)& :=\{y\in {\mathbb{R}}_{+}^2|y_1^2+y_2^2\ge x^2\}\cup \left(\right[-1,0]\times [|x|,\mathrm{\infty }\left)\right).\end{array}$A simple calculation reveals $\begin{array}{rl}\mathrm{\forall x}\in \mathbb{R}:\mathrm{\Psi }\left(x\right)& =\{\begin{array}{ll}\left\{\right(|x|\cos t,|x|\sin t)|t\in [0,\pi /2\left)\right\}\cup \left\{\right(-1,|x|\left)\right\}& x\ne 0,\\ \left\{\right(-1,0\left)\right\}& x=0,\end{array}\\ {\mathrm{\Psi }}_{\mathrm{w}}\left(\mathrm{x}\right)& =\left\{\right(|x|\cos t,|x|\sin t)|t\in [0,\pi /2\left]\right\}\cup \left(\right[-1,0]\times \{|x|\left\}\right)\\ & \cup \left(\right[|x|,\mathrm{\infty })\times \{0\left\}\right)\cup \left(\right\{-1\}\times [|x|,\mathrm{\infty }\left)\right),\\ \overline{\mathrm{\Psi }}\left(x\right)& =\left\{\right(|x|\cos t,|x|\sin t)|t\in [0,\pi /2\left]\right\}\cup \left\{\right(-1,|x|\left)\right\}.\end{array}$Observe that $\mathrm{gph}\overline{\mathrm{\Psi }}=\mathrm{gph}\mathrm{\Psi }\cup \left\{\right(x,(0,|x|))|x\in \mathbb{R}\}$ holds true.

Now, we consider the superordinate semivectorial bilevel optimization problem $\underset{x,y}{min}\left\{\right(y_1+1/4{)}^2+y_2^2|x\ge 1,y\in \hat{\mathrm{\Psi }}\left(x\right)\}$where $\hat{\mathrm{\Psi }}\in \{\mathrm{\Psi },{\mathrm{\Psi }}_{\mathrm{w}},\overline{\mathrm{\Psi }}\}$. Let us start with the investigation of $\hat{\mathrm{\Psi }}:=\mathrm{\Psi }$. The associated problem ($\mathcal{E}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ) does not possess a global minimizer, but there is a local minimizer at $(1,(-1,1\left)\right)$. Whenever $\hat{\mathrm{\Psi }}:={\mathrm{\Psi }}_{\mathrm{w}}$, the associated problem (${\mathcal{E}}_{\mathrm{w}}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ) possesses the uniquely determined global minimizer $(1,(-1/4,1\left)\right)$, and no other local minimizers exist. Finally, for $\hat{\mathrm{\Psi }}:=\overline{\mathrm{\Psi }}$, the associated problem ($\overline{\mathcal{E}}$-BPP(15) $D^{\ast }\mathrm{\Phi }(\overline{x},\overline{z})\left(z^{\ast }\right)\subset D^{\ast }\mathrm{\Sigma }(\overline{x},\overline{z})\left(z^{\ast }\right)$(15) ) possesses the uniquely determines global minimizer $(1,(0,1\left)\right)$ (which is not feasible to ($\mathcal{E}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) )) and an additional local minimizer at $(1,(-1,1\left)\right)$.

Finally, we would like to point the reader's attention to the fact that, in contrast to Corollary 3.4, ($\overline{\mathcal{E}}$-VFR(14) $D^{\ast }\mathrm{\Phi }(\overline{x},\overline{z})\left(z^{\ast }\right)\subset D^{\ast }(\mathrm{\Sigma }+\mathcal{C})(\overline{x},\overline{z})\left(z^{\ast }\right)$(14) ) and (13) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \overline{\mathrm{\Phi }}(x)-\mathcal{C},y\in \mathrm{\Gamma }(x\left)\right\}$(13) are not necessarily equivalent as the feasible set of (13(16) $D^{\ast }{\mathrm{\Phi }}_{\mathrm{w}}(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)\subset {\mathrm{D}}^{\ast }(\mathrm{\Sigma }+\mathcal{C})(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)$(16) ) is likely to be larger than the one of ($\overline{\mathcal{E}}$-VFR(14) $D^{\ast }\mathrm{\Phi }(\overline{x},\overline{z})\left(z^{\ast }\right)\subset D^{\ast }(\mathrm{\Sigma }+\mathcal{C})(\overline{x},\overline{z})\left(z^{\ast }\right)$(14) ). However, if $\mathrm{gph}{\mathrm{\Phi }}_{\mathrm{w}}$ is closed, the feasible set of (13(16) $D^{\ast }{\mathrm{\Phi }}_{\mathrm{w}}(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)\subset {\mathrm{D}}^{\ast }(\mathrm{\Sigma }+\mathcal{C})(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)$(16) ) is included in the feasible set of (6b(8) $\mathrm{\forall x}\in {\mathbb{R}}^n\mathrm{\forall y}\in {\mathbb{R}}^m:f(x,y):=\mathrm{Dy},\mathrm{\Gamma }\left(x\right):=\{y|\mathrm{Ax}+\mathrm{By}\le d\}.$(8) ) which, by Corollary 3.4, equals the feasible set of (${\mathcal{E}}_{\mathrm{w}}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ). Thus, the enlargement of the feasible set in (13(16) $D^{\ast }{\mathrm{\Phi }}_{\mathrm{w}}(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)\subset {\mathrm{D}}^{\ast }(\mathrm{\Sigma }+\mathcal{C})(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)$(16) ) in comparison with ($\overline{\mathcal{E}}$-VFR(14) $D^{\ast }\mathrm{\Phi }(\overline{x},\overline{z})\left(z^{\ast }\right)\subset D^{\ast }(\mathrm{\Sigma }+\mathcal{C})(\overline{x},\overline{z})\left(z^{\ast }\right)$(14) ) is somewhat controllable.

Example 3.21

Let us reinspect the semivectorial bilevel optimization problem from Example 3.20. One can easily check that, for $\overline{x}:=1$, $\overline{y}:=(-1/4,1)\in \mathrm{\Gamma }\left(\overline{x}\right)$ satisfies $f(\overline{x},\overline{y})\in \overline{\mathrm{\Phi }}\left(\overline{x}\right)-{\mathbb{R}}_{+}^2=\overline{\mathrm{\Psi }}\left(\overline{x}\right)-{\mathbb{R}}_{+}^2,$but $(\overline{x},\overline{y})$ is not feasible to the associated problem ($\overline{\mathcal{E}}$-BPP(15) $D^{\ast }\mathrm{\Phi }(\overline{x},\overline{z})\left(z^{\ast }\right)\subset D^{\ast }\mathrm{\Sigma }(\overline{x},\overline{z})\left(z^{\ast }\right)$(15) ) and, thus, ($\overline{\mathcal{E}}$-VFR(14) $D^{\ast }\mathrm{\Phi }(\overline{x},\overline{z})\left(z^{\ast }\right)\subset D^{\ast }(\mathrm{\Sigma }+\mathcal{C})(\overline{x},\overline{z})\left(z^{\ast }\right)$(14) ). Keeping our above discussion in mind and noticing that $\mathrm{gph}{\mathrm{\Phi }}_{\mathrm{w}}$ is closed, the associated model (13(16) $D^{\ast }{\mathrm{\Phi }}_{\mathrm{w}}(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)\subset {\mathrm{D}}^{\ast }(\mathrm{\Sigma }+\mathcal{C})(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)$(16) ) possesses the global minimizer $(\overline{x},\overline{y})$, see Example 3.20 as well.

4. Towards necessary optimality conditions

For the derivation of necessary optimality conditions and constraint qualifications for ($\mathcal{E}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ) and (${\mathcal{E}}_{\mathrm{w}}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ) via ($\mathcal{E}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ) and (${\mathcal{E}}_{\mathrm{w}}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ), respectively, in terms of the limiting tools of generalized differentiation, it is important to have formulas or at least upper estimates for the coderivative of Φ and ${\mathrm{\Phi }}_{\mathrm{w}}$ available. Here, we focus on the latter and comment on two approaches which can be used to obtain such estimates in terms of initial problem data.

4.1. Coderivative estimates via the domination property

In the literature, one can find several papers dealing with the derivation of upper coderivative estimates for the frontier mapping Φ in the case where (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ) possesses the domination property for all $x\in {\mathbb{R}}^n$ from a neighborhood of the reference point, see e.g. [40–42]. Apart from the fact that these papers completely ignore the fact that $\mathrm{gph}\mathrm{\Phi }$ is likely to be non-closed, so that the coderivative is not well-defined, they obtain validity of the upper estimate (14) $D^{\ast }\mathrm{\Phi }(\overline{x},\overline{z})\left(z^{\ast }\right)\subset D^{\ast }(\mathrm{\Sigma }+\mathcal{C})(\overline{x},\overline{z})\left(z^{\ast }\right)$(14) only for points $(\overline{x},\overline{z})\in \mathrm{gph}\mathrm{\Phi }$ and $z^{\ast }\in {\mathcal{C}}_{>}^{\ast }$ where ${\mathcal{C}}_{>}^{\ast }:=\{z^{\ast }|\mathrm{\forall z}\in \mathcal{C}\cap {\mathbb{S}}_1(0):〈z^{\ast },z〉>0\}.$In practice, this means that the estimate (14(17) $(x,z)\mapsto \left\{\right(x,y)|y\in \mathrm{\Gamma }(x),f(x,y)=z\}$(17) ) is, more or less, only reasonable for arguments $z^{\ast }\in {\mathbb{R}}^q$ from $\mathrm{int}{\mathcal{C}}^{\ast }$. For the derivation of necessary optimality conditions for ($\mathcal{E}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ) via ($\mathcal{E}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ), however, this additional constraint on the variable $z^{\ast }$ might be too restrictive.

In the proof of [24, Theorem 4.1], the authors claim that for $\mathcal{C}:={\mathbb{R}}_{+}^q$, one has the estimate (15) $D^{\ast }\mathrm{\Phi }(\overline{x},\overline{z})\left(z^{\ast }\right)\subset D^{\ast }\mathrm{\Sigma }(\overline{x},\overline{z})\left(z^{\ast }\right)$(15) for all $z^{\ast }\in {\mathbb{R}}^q$. This estimate, however, does not depend on the choice of the ordering cone $\mathcal{C}$ which is somewhat dubious. As the following example shows, this estimate is indeed false, and so are [24, Theorems 4.1, 4.4].

Example 4.1

We consider (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ) with $n:=1,m:=q:=2$, $\mathcal{C}:={\mathbb{R}}_{+}^2$, and $\begin{array}{rl}\mathrm{\forall x}\in \mathbb{R}\mathrm{\forall y}\in {\mathbb{R}}^2:f(x,y)& :=y,\\ \mathrm{\Gamma }\left(x\right)& :=\{y|y_1+2y_2\ge 0,2y_1+y_2\ge 0,y_1,y_2\le 2\},\end{array}$i.e. the associated problem (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ) does not depend on the parameter. We find $\begin{array}{rl}\mathrm{\forall x}\in \mathbb{R}:\mathrm{\Sigma }\left(x\right)& =\mathrm{\Gamma }\left(x\right),\\ \mathrm{\Phi }\left(x\right)& =\mathrm{\Psi }\left(x\right)=\mathrm{conv}\left\{\right(-1,2),(0,0\left)\right\}\cup \mathrm{conv}\left\{\right(2,-1),(0,0\left)\right\}\end{array}$and observe that (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ) possesses the domination property (for each $x\in \mathbb{R}$). Let us fix $\overline{x}:=0$ and $\overline{z}:=(0,0)$. Then we find $\begin{array}{rl}{N}_{\mathrm{gph}\mathrm{\Sigma }}(\overline{x},\overline{z})& =\mathrm{cone}\left\{\right(0,-1,-2),(0,-2,-1\left)\right\},\\ N_{\mathrm{gph}\mathrm{\Phi }}(\overline{x},\overline{z})& =\mathrm{cone}\left\{\right(0,-1,-2),(0,-2,-1\left)\right\}\\ & \cup \mathrm{cone}\left\{\right(0,2,1\left)\right\}\cup \mathrm{cone}\left\{\right(0,1,2\left)\right\}.\end{array}$This can be easily seen from $\mathrm{gph}\mathrm{\Sigma }=\mathbb{R}\times \mathrm{\Sigma }\left(0\right)$ and $\mathrm{gph}\mathrm{\Phi }=\mathbb{R}\times \mathrm{\Phi }\left(0\right)$, the product rule for the computation of limiting normals, see e.g. [38, Proposition 1.2], and a simple calculation of the normal cones to $\mathrm{\Sigma }\left(0\right)$ (which is convex) and $\mathrm{\Phi }\left(0\right)$ in ${\mathbb{R}}^2$. Hence, for $z^{\ast }:=(-1,-2)$, we find $D^{\ast }\mathrm{\Phi }(\overline{x},\overline{z})\left(z^{\ast }\right)=\left\{0\right\},D^{\ast }\mathrm{\Sigma }(\overline{x},\overline{z})\left(z^{\ast }\right)=\mathrm{\varnothing },$showing that the estimate (15(P(x,λ)) $\underset{y}{min}\{〈\lambda ,f(x,y)〉|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x,λ)) ) fails to be true for general arguments.

Our next example indicates that the results from [40–42] can be directly generalized to weak frontier mappings.

Example 4.2

We consider (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ) with $n:=m:=1,q:=2$, $\mathcal{C}:={\mathbb{R}}_{+}^2$, and $\mathrm{\forall x},y\in \mathbb{R}:f(x,y):=(\mathrm{xy},y),\mathrm{\Gamma }\left(x\right):=[0,1].$Then we find $\mathrm{\Sigma }\left(x\right)=\mathrm{conv}\left\{\right(0,0),(x,1\left)\right\}$ for all $x\in \mathbb{R}$ as well as $\mathrm{\forall x}\in \mathbb{R}:{\mathrm{\Phi }}_{\mathrm{w}}\left(\mathrm{x}\right)=\{\begin{array}{ll}\mathrm{\Sigma }\left(x\right)& x\le 0,\\ \left\{\right(0,0\left)\right\}& x>0,\end{array}{\mathrm{\Psi }}_{\mathrm{w}}\left(\mathrm{x}\right)=\{\begin{array}{ll}[0,1]& x\le 0,\\ \left\{0\right\}& x>0.\end{array}$Using this, one can easily check that $\mathrm{\Sigma }\left(x\right)\subset {\mathrm{\Phi }}_{\mathrm{w}}\left(\mathrm{x}\right)+{\mathbb{R}}_{+}^2$ holds for all $x\in \mathbb{R}$, i.e. the associated problem (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ) possesses the weak domination property for all $x\in \mathbb{R}$ (actually, it also possesses the domination property for all $x\in \mathbb{R}$). Let us consider $\overline{x}:=\overline{y}:=0$ and $\overline{z}:=(0,0)$. It is not hard to see that $\begin{array}{rl}{N}_{\mathrm{gph}\mathrm{\Sigma }}(\overline{x},\overline{z})& =\left\{0\right\}\times \mathbb{R}\times {\mathbb{R}}_{-},\\ N_{\mathrm{gph}(\mathrm{\Sigma }+\mathcal{C})}(\overline{x},\overline{z})& =\left\{0\right\}\times {\mathbb{R}}_{-}\times {\mathbb{R}}_{-},\\ N_{\mathrm{gph}{\mathrm{\Phi }}_{\mathrm{w}}}(\overline{x},\overline{z})& =\left(\right\{0\}\times \mathbb{R}\times \mathbb{R})\cup ({\mathbb{R}}_{+}\times \mathbb{R}\times \{0\left\}\right),\\ N_{\mathrm{gph}{\mathrm{\Psi }}_{\mathrm{w}}}(\overline{x},\overline{y})& =\left(\right\{0\}\times \mathbb{R})\cup ({\mathbb{R}}_{+}\times \{0\left\}\right).\end{array}$This directly shows that the inclusion (16) $D^{\ast }{\mathrm{\Phi }}_{\mathrm{w}}(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)\subset {\mathrm{D}}^{\ast }(\mathrm{\Sigma }+\mathcal{C})(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)$(16) holds for all $z^{\ast }\in \mathrm{int}{\mathbb{R}}_{+}^2$, but is violated for, e.g. $z^{\ast }:=(1,0)$ already, where $D^{\ast }{\mathrm{\Phi }}_{\mathrm{w}}(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)={\mathbb{R}}_{+},{\mathrm{D}}^{\ast }(\mathrm{\Sigma }+\mathcal{C})(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)=\left\{0\right\}.$

Let us now formally prove that the estimate (16(16) $D^{\ast }{\mathrm{\Phi }}_{\mathrm{w}}(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)\subset {\mathrm{D}}^{\ast }(\mathrm{\Sigma }+\mathcal{C})(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)$(16) ) holds for all $z^{\ast }\in {\mathcal{C}}_{>}^{\ast }$.

Lemma 4.3

Let $(\overline{x},\overline{z})\in \mathrm{gph}{\mathrm{\Phi }}_{\mathrm{w}}$ be fixed and let $\mathrm{gph}{\mathrm{\Phi }}_{\mathrm{w}}$ and $\mathrm{gph}(\mathrm{\Sigma }+\mathcal{C})$ be locally closed around $(\overline{x},\overline{z})$. Assume that (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ) possesses the weak domination property for all $x\in U$ where $U\subset {\mathbb{R}}^n$ is a neighborhood of $\overline{x}$. Then, for each $z^{\ast }\in {\mathcal{C}}_{>}^{\ast }$, the estimate (16(16) $D^{\ast }{\mathrm{\Phi }}_{\mathrm{w}}(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)\subset {\mathrm{D}}^{\ast }(\mathrm{\Sigma }+\mathcal{C})(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)$(16) ) is valid.

Proof.

Fix $z^{\ast }\in {\mathcal{C}}_{>}^{\ast }$. By mimicking the proof of [40, Lemma 3.2], which does not rely on the precise underlying notion of nondominance, we find $D^{\ast }{\mathrm{\Phi }}_{\mathrm{w}}(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)\subset {\mathrm{D}}^{\ast }({\mathrm{\Phi }}_{\mathrm{w}}+\mathcal{C})(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right).$Furthermore, locally around $\overline{x}$, the mappings ${\mathrm{\Phi }}_{\mathrm{w}}+\mathcal{C}$ and $\mathrm{\Sigma }+\mathcal{C}$ are identical by the weak domination property. Noting that the coderivative is a local construction, the claim follows.

We would like to point the reader's attention to the similarities in the estimates (14(17) $(x,z)\mapsto \left\{\right(x,y)|y\in \mathrm{\Gamma }(x),f(x,y)=z\}$(17) ) and (16(16) $D^{\ast }{\mathrm{\Phi }}_{\mathrm{w}}(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)\subset {\mathrm{D}}^{\ast }(\mathrm{\Sigma }+\mathcal{C})(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)$(16) ). Indeed, the right-hand side is the same in both estimates which is somewhat unsatisfying as the precise underlying notion of nondominance seems to make no difference. More precisely, it does not seem to be too reasonable to try to find assumptions which guarantee correctness of the results, e.g. in [40] for the frontier mapping as we obtained the same upper estimate for the coderivative of the weak frontier mapping under most likely milder conditions.

With the aid of Lemma 4.3, we are in position to obtain an upper estimate for the coderivative of ${\mathrm{\Phi }}_{\mathrm{w}}$ in terms of initial problem data. The following result is inspired by [40, Theorem 3.3].

Theorem 4.4

Let $(\overline{x},\overline{z})\in \mathrm{gph}{\mathrm{\Phi }}_{\mathrm{w}}$ be fixed and let $\mathrm{gph}{\mathrm{\Phi }}_{\mathrm{w}}$ be locally closed around $(\overline{x},\overline{z})$. Assume that (P(x)(P(x)) $\underset{y}{min}{}_{\mathcal{C}}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x)) ) possesses the weak domination property for all $x\in U$ where $U\subset {\mathbb{R}}^n$ is a neighborhood of $\overline{x}$. Furthermore, let Γ be locally bounded at $\overline{x}$. Let f be locally Lipschitz continuous at all points $(\overline{x},\overline{y})$ such that $\overline{y}\in {\mathrm{\Psi }}_{\mathrm{w}}\left(\overline{\mathrm{x}}\right)$ and $f(\overline{x},\overline{y})=\overline{z}$ hold. Then, for each $z^{\ast }\in {\mathcal{C}}_{>}^{\ast }$, $D^{\ast }{\mathrm{\Phi }}_{\mathrm{w}}(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)\subset \{{\mathrm{x}}_1^{\ast }+{\mathrm{x}}_2^{\ast }|\begin{array}{rl}& \overline{y}\in {\mathrm{\Psi }}_{\mathrm{w}}\left(\overline{\mathrm{x}}\right),\mathrm{f}(\overline{\mathrm{x}},\overline{\mathrm{y}})=\overline{\mathrm{z}},\\ & (x_1^{\ast },y^{\ast })\in D^{\ast }f(\overline{x},\overline{y})\left(z^{\ast }\right),x_2^{\ast }\in D^{\ast }\mathrm{\Gamma }(\overline{x},\overline{y})\left(y^{\ast }\right)\end{array}\}.$If, additionally, f is continuously differentiable at all points $(\overline{x},\overline{y})$ such that $\overline{y}\in {\mathrm{\Psi }}_{\mathrm{w}}\left(\overline{\mathrm{x}}\right)$ and $f(\overline{x},\overline{y})=\overline{z}$ hold, then, for each $z^{\ast }\in {\mathcal{C}}_{>}^{\ast }$, $D^{\ast }{\mathrm{\Phi }}_{\mathrm{w}}(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)\subset \left\{{\mathrm{f}}_{\mathrm{x}}^{\prime }\right(\overline{\mathrm{x}},\overline{\mathrm{y}}{)}^{\mathrm{\top }}{\mathrm{z}}^{\ast }+{\mathrm{x}}^{\ast }\left|\begin{array}{rl}& \overline{y}\in {\mathrm{\Psi }}_{\mathrm{w}}\left(\overline{\mathrm{x}}\right),\mathrm{f}(\overline{\mathrm{x}},\overline{\mathrm{y}})=\overline{\mathrm{z}},\\ & x^{\ast }\in D^{\ast }\mathrm{\Gamma }(\overline{x},\overline{y})(f_y^{\prime }(\overline{x},\overline{y}{)}^{\mathrm{\top }}{z}^{\ast })\end{array}\right\}.$

Proof.

Let us note that closedness of $\mathrm{gph}\mathrm{\Gamma }$ and local boundedness of Γ at $\overline{x}$ yield that, locally around $(\overline{x},\overline{z})$, $\mathrm{gph}\mathrm{\Sigma }$ and $\mathrm{gph}(\mathrm{\Sigma }+\mathcal{C})$ are closed. Combining Lemma 4.3 and the first part of the proof of [40, Theorem 3.3] while taking into account that $-N_{\mathcal{C}}\left(0\right)={\mathcal{C}}^{\ast }\supset {\mathcal{C}}_{>}^{\ast }$ since $\mathcal{C}$ is a cone gives $D^{\ast }{\mathrm{\Phi }}_{\mathrm{w}}(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)\subset {\mathrm{D}}^{\ast }\mathrm{\Sigma }(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)$for each $z^{\ast }\in {\mathcal{C}}_{>}^{\ast }$.

Defining $\tilde{\mathrm{\Gamma }}:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^n\times {\mathbb{R}}^m$ by $\tilde{\mathrm{\Gamma }}\left(x\right):=\left\{x\right\}\times \mathrm{\Gamma }\left(x\right)$ for each $x\in {\mathbb{R}}^n$ gives $\mathrm{\Sigma }=f\circ \tilde{\mathrm{\Gamma }}$, and $D^{\ast }\tilde{\mathrm{\Gamma }}(\overline{x},(\overline{x},y\left)\right)(x^{\ast },y^{\ast })=\left\{x^{\ast }\right\}+D^{\ast }\mathrm{\Gamma }(\overline{x},y)\left(y^{\ast }\right)$holds for each $y\in \mathrm{\Gamma }\left(\overline{x}\right)$ and $(x^{\ast },y^{\ast })\in {\mathbb{R}}^n\times {\mathbb{R}}^m$, see e.g. [46, Lemma 5.7]. Thus, in order to infer the more general assertion of the theorem, we apply the chain rule from [46, Theorem 5.2]. Therefore, we note that the involved intermediate mapping (17) $(x,z)\mapsto \left\{\right(x,y)|y\in \mathrm{\Gamma }(x),f(x,y)=z\}$(17) is inner semicompact at $(\overline{x},\overline{z})$ w.r.t. its domain since Γ is assumed to be locally bounded at $\overline{x}$, and that f is Lipschitz continuous at $(\overline{x},\overline{y})$ which serves as a qualification condition. Then we find $D^{\ast }\mathrm{\Sigma }(\overline{x},\overline{z})\left(z^{\ast }\right)\subset \left\{{\tilde{x}}^{\ast }\right|\begin{array}{rl}& \overline{y}\in \mathrm{\Gamma }\left(\overline{x}\right),f(\overline{x},\overline{y})=\overline{z},\\ & (x^{\ast },y^{\ast })\in D^{\ast }f(\overline{x},\overline{y})\left(z^{\ast }\right),{\tilde{x}}^{\ast }\in D^{\ast }\tilde{\mathrm{\Gamma }}(\overline{x},(\overline{x},\overline{y}\left)\right)(x^{\ast },y^{\ast })\end{array}\},$and noting that, due to Proposition 3.1, each $\overline{y}\in \mathrm{\Gamma }\left(\overline{x}\right)$ with $f(\overline{x},\overline{y})=\overline{z}$ already belongs to ${\mathrm{\Psi }}_{\mathrm{w}}\left(\overline{\mathrm{x}}\right)$ yields the claim.

The second statement of the theorem is an obvious consequence of the first assertion.

Let us note that the chain rule from [46, Theorem 5.2] also allows for the derivation of refined upper estimates in terms of directional coderivatives.

We would like to point the interested reader to the fact that in the proofs of [40, Theorem 3.3] and [24, Theorem 4.1], the authors rely on the chain rule from [38, Theorem 3.18(i)] which is only possible if the intermediate mapping from (17(17) $(x,z)\mapsto \left\{\right(x,y)|y\in \mathrm{\Gamma }(x),f(x,y)=z\}$(17) ) possesses certain inner semicontinuity properties. These, however, are not inherent from the assumptions stated in [24,40], and one can easily check that even inner semicontinuity of Γ is not sufficient for these requirements. Thus, the aforementioned results are not reliable. Let us also note that, in situations where f is Lipschitzian, the coderivative of Σ can be computed exploiting a combination of the image and pre-image rule as we have $\mathrm{gph}\mathrm{\Sigma }=\mathrm{\Pi }\left(\right\{(x,y,z)|(x,y,f(x,y)-z)\in \mathrm{gph}\mathrm{\Gamma }\times \left\{0\right\}\left\}\right)$where $\mathrm{\Pi }:{\mathbb{R}}^n\times {\mathbb{R}}^m\times {\mathbb{R}}^q\to {\mathbb{R}}^n\times {\mathbb{R}}^q$ is given by $\mathrm{\Pi }(x,y,z):=(x,z)$ for all $x\in {\mathbb{R}}^n$, $y\in {\mathbb{R}}^m$, and $z\in {\mathbb{R}}^q$, see e.g. [46, Sections 5.1.3, 5.1.4]. This, however, still makes some additional boundedness assumption on Γ necessary, and the resulting upper estimate on the coderivative of Σ would also comprise a union over ${\mathrm{\Psi }}_{\mathrm{w}}\left(\overline{\mathrm{x}}\right)$. More precisely, this approach precisely recovers the estimates from Theorem 4.4.

Let us specify Theorem 4.4 for the scalar case where $q:=1$ and $\mathcal{C}:={\mathbb{R}}_{+}$. Then the weak domination property is inherently satisfied. Furthermore, we find $1\in {\mathcal{C}}_{>}^{\ast }=(0,\mathrm{\infty })$, and $D^{\ast }{\mathrm{\Phi }}_{\mathrm{w}}(\overline{\mathrm{x}},\phi (\overline{\mathrm{x}}\left)\right)\left(1\right)$ equals the so-called limiting subdifferential at $\overline{x}$ of the optimal value function φ from (1(1) $\mathrm{\forall x}\in {\mathbb{R}}^n:\phi \left(x\right):=\underset{y}{inf}\left\{f\right(x,y)|y\in \mathrm{\Gamma }(x\left)\right\},$(1) ) provided the latter is continuous around $\overline{x}$, see [38, Definition 1.77, Theorem 1.80]. Then the estimates in Theorem 4.4 recover those obtained in [54, Theorem 7] under slightly different assumptions. Hence, one can interpret Theorem 4.4 as a generalization of this result to the multiobjective situation. Nevertheless, Theorem 4.4 is of limited practical use as it provides no information about situations where the argument $z^{\ast }$ of the coderivative is chosen in ${\mathbb{R}}^q\setminus {\mathcal{C}}_{>}^{\ast }$, and such values of $z^{\ast }$ might be of interest when deriving necessary optimality conditions for (${\mathcal{E}}_{\mathrm{w}}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ) as the concept of subdifferentials is not reasonable in vector-valued scenarios whenever scalarization is not taken into account.

4.2. Coderivative estimates via scalarization

In this section, we aim to find upper coderivative estimates for the weak frontier mapping ${\mathrm{\Phi }}_{\mathrm{w}}$ via a scalarization approach. This, however, makes additional assumptions necessary.

Assumption 4.5

For each $x\in {\mathbb{R}}^n$, $f(x,\cdot )$ is $\mathcal{C}$-convex and $\mathrm{\Gamma }\left(x\right)$ is convex.

We emphasize that Assumption 4.5 guarantees that $\mathrm{\Sigma }\left(x\right)+\mathcal{C}$ is convex for each $x\in {\mathbb{R}}^n$. Hence, due to [2, Corollary 5.29], for each $\overline{x}\in \mathrm{dom}{\mathrm{\Psi }}_{\mathrm{w}}$, we find the equivalence $y\in {\mathrm{\Psi }}_{\mathrm{w}}\left(\overline{\mathrm{x}}\right)⟺\mathrm{\exists \lambda }\in {\mathcal{C}}^{\ast }\cap {\mathbb{S}}_1\left(0\right):\mathrm{y}\in {\mathrm{\Psi }}_{\mathrm{w}}^{\mathrm{sc}}(\overline{\mathrm{x}},\lambda )$where ${\mathrm{\Psi }}_{\mathrm{w}}^{\mathrm{sc}}:{\mathbb{R}}^{\mathrm{n}}\times {\mathbb{R}}^{\mathrm{q}}\rightrightarrows {\mathbb{R}}^{\mathrm{m}}$ is the solution mapping associated with the scalar parametric optimization problem (P(x,λ)) $\underset{y}{min}\{〈\lambda ,f(x,y)〉|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x,λ)) Based on the chain rule, we obtain a simple upper estimate for the coderivative of ${\mathrm{\Phi }}_{\mathrm{w}}$ in terms of the coderivative of ${\mathrm{\Psi }}_{\mathrm{w}}$, as we have ${\mathrm{\Phi }}_{\mathrm{w}}=\mathrm{f}\circ {\tilde{\mathrm{\Psi }}}_{\mathrm{w}}$ where the mapping ${\tilde{\mathrm{\Psi }}}_{\mathrm{w}}:{\mathbb{R}}^{\mathrm{n}}\rightrightarrows {\mathbb{R}}^{\mathrm{n}}\times {\mathbb{R}}^{\mathrm{m}}$ is given by ${\tilde{\mathrm{\Psi }}}_{\mathrm{w}}\left(\mathrm{x}\right):=\left\{\mathrm{x}\right\}\times {\mathrm{\Psi }}_{\mathrm{w}}\left(\mathrm{x}\right)$ for each $x\in {\mathbb{R}}^n$. We omit the proof as it mainly parallels the arguments used to validate Theorem 4.4.

Lemma 4.6

Fix $(\overline{x},\overline{z})\in \mathrm{gph}{\mathrm{\Phi }}_{\mathrm{w}}$ and let $\mathrm{gph}{\mathrm{\Phi }}_{\mathrm{w}}$ be locally closed around $(\overline{x},\overline{z})$. Let ${\mathrm{\Psi }}_{\mathrm{w}}$ be inner semicompact at $\overline{x}$ w.r.t. $\mathrm{dom}{\mathrm{\Psi }}_{\mathrm{w}}$. Furthermore, for each $\overline{y}\in {\mathrm{\Psi }}_{\mathrm{w}}\left(\overline{\mathrm{x}}\right)$ with $f(\overline{x},\overline{y})=\overline{z}$, let $\mathrm{gph}{\mathrm{\Psi }}_{\mathrm{w}}$ be closed locally around $(\overline{x},\overline{y})$, and let f be locally Lipschitz continuous at $(\overline{x},\overline{y})$. Then, for each $z^{\ast }\in {\mathbb{R}}^q$, $D^{\ast }{\mathrm{\Phi }}_{\mathrm{w}}(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)\subset \{{\mathrm{x}}_1^{\ast }+{\mathrm{x}}_2^{\ast }|\begin{array}{rl}& \overline{y}\in {\mathrm{\Psi }}_{\mathrm{w}}\left(\overline{\mathrm{x}}\right),\mathrm{f}(\overline{\mathrm{x}},\overline{\mathrm{y}})=\overline{\mathrm{z}},\\ & (x_1^{\ast },y^{\ast })\in D^{\ast }f(\overline{x},\overline{y})\left(z^{\ast }\right),x_2^{\ast }\in D^{\ast }{\mathrm{\Psi }}_{\mathrm{w}}(\overline{\mathrm{x}},\overline{\mathrm{y}})\left({\mathrm{y}}^{\ast }\right)\end{array}\}.$If, additionally, f is continuously differentiable at all points $(\overline{x},\overline{y})$ such that $\overline{y}\in {\mathrm{\Psi }}_{\mathrm{w}}\left(\overline{\mathrm{x}}\right)$ and $f(\overline{x},\overline{y})=\overline{z}$ hold, then, for each $z^{\ast }\in {\mathbb{R}}^q$, $D^{\ast }{\mathrm{\Phi }}_{\mathrm{w}}(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)\subset \left\{{\mathrm{f}}_{\mathrm{x}}^{\prime }\right(\overline{\mathrm{x}},\overline{\mathrm{y}}{)}^{\mathrm{\top }}{\mathrm{z}}^{\ast }+{\mathrm{x}}^{\ast }\left|\begin{array}{rl}& \overline{y}\in {\mathrm{\Psi }}_{\mathrm{w}}\left(\overline{\mathrm{x}}\right),\mathrm{f}(\overline{\mathrm{x}},\overline{\mathrm{y}})=\overline{\mathrm{z}},\\ & x^{\ast }\in D^{\ast }{\mathrm{\Psi }}_{\mathrm{w}}(\overline{\mathrm{x}},\overline{\mathrm{y}})({\mathrm{f}}_{\mathrm{y}}^{\prime }(\overline{\mathrm{x}},\overline{\mathrm{y}}{)}^{\mathrm{\top }}{\mathrm{z}}^{\ast })\end{array}\right\}.$

Let us emphasize that Lemma 4.6 is correct even in the absence of Assumption 4.5 which is not needed for the proof. We note that the estimates from Lemma 4.6 are closely related to those ones obtained in Theorem 4.4. Indeed, the only difference (apart from the underlying assumptions) is that in Theorem 4.4, all estimates are valid for $z^{\ast }\in {\mathcal{C}}_{>}^{\ast }$ only, while the coderivative of the efficiency mapping ${\mathrm{\Psi }}_{\mathrm{w}}$ appearing in Lemma 4.6 is replaced by the coderivative of the feasibility mapping Γ in Theorem 4.4. Let us also visualize this in exemplary way.

One can easily check that the estimate provided in Lemma 4.6 is actually sharp in the context of Example 4.2. Indeed, one obtains $D^{\ast }{\mathrm{\Phi }}_{\mathrm{w}}(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)={\mathrm{D}}^{\ast }{\mathrm{\Psi }}_{\mathrm{w}}(\overline{\mathrm{x}},\overline{\mathrm{y}})\left({\mathrm{z}}_2^{\ast }\right)=\{\begin{array}{ll}\left\{0\right\}& z_2^{\ast }\ne 0,\\ {\mathbb{R}}_{+}& z_2^{\ast }=0\end{array}$for each $z^{\ast }\in {\mathbb{R}}^2$ in this situation (observe that the only point $y\in {\mathrm{\Psi }}_{\mathrm{w}}\left(\overline{\mathrm{x}}\right)$ which satisfies $f(\overline{x},y)=\overline{z}$ in the present situation is $\overline{y}$). We also note that the estimates from Theorem 4.4 do not apply for $z^{\ast }\in \mathbb{R}\times \left\{0\right\}$ as these vectors do not belong to $\mathrm{int}{\mathbb{R}}_{+}^2$.

It is an essential disadvantage of the estimates from Lemma 4.6 that they are not stated in terms of initial problem data as the coderivative of ${\mathrm{\Psi }}_{\mathrm{w}}$ is a fully implicit object. Subsequently, we aim to estimate it from above. In order to deal with this issue, Assumption 4.5 turns out to be helpful as we will demonstrate in the results below. For brevity of notation, we introduce an intermediate mapping $\mathrm{\Xi }:{\mathbb{R}}^n\times {\mathbb{R}}^m\rightrightarrows {\mathbb{R}}^q$ given by $\mathrm{\forall x}\in {\mathbb{R}}^n\mathrm{\forall y}\in {\mathbb{R}}^m:\mathrm{\Xi }(x,y):=\{\lambda \in {\mathcal{C}}^{\ast }\cap {\mathbb{S}}_1(0)|y\in {\mathrm{\Psi }}_{\mathrm{w}}^{\mathrm{sc}}(\mathrm{x},\lambda \left)\right\}.$For given $(x,y)\in \mathrm{gph}{\mathrm{\Psi }}_{\mathrm{w}}$, $\mathrm{\Xi }(x,y)$ comprises all (normalized) scalarization parameters for which y is a minimizer of (P$(x,\lambda )$(P(x,λ)) $\underset{y}{min}\{〈\lambda ,f(x,y)〉|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x,λ)) ). We note that, by compactness of ${\mathcal{C}}^{\ast }\cap {\mathbb{S}}_1\left(0\right)$, Ξ is inner semicompact w.r.t. its domain at each point of its domain.

Lemma 4.7

Fix $(\overline{x},\overline{y})\in \mathrm{gph}{\mathrm{\Psi }}_{\mathrm{w}}$ and let $\mathrm{gph}{\mathrm{\Psi }}_{\mathrm{w}}$ be closed locally around $(\overline{x},\overline{y})$. Furthermore, for each $\lambda \in \mathrm{\Xi }(\overline{x},\overline{y})$, let $\mathrm{gph}{\mathrm{\Psi }}_{\mathrm{w}}^{\mathrm{sc}}$ be closed locally around $\left(\right(\overline{x},\lambda ),\overline{y})$. Additionally, let Assumption 4.5 hold. Finally, let one of the following conditions be valid:

(a)	${\mathrm{\Psi }}_{\mathrm{w}}^{\mathrm{sc}}$ is a polyhedral set-valued mapping, $\mathcal{C}$ is a polyhedral cone, and the unit sphere ${\mathbb{S}}_1\left(0\right)$ is taken w.r.t. the 1- or ∞-norm,
(b)	for each $\lambda \in \mathrm{\Xi }(\overline{x},\overline{y})$, the qualification condition $(0,\eta )\in D^{\ast }{\mathrm{\Psi }}_{\mathrm{w}}^{\mathrm{sc}}\left(\right(\overline{\mathrm{x}},\lambda ),\overline{\mathrm{y}})\left(0\right),\eta \in -{\mathrm{N}}_{{\mathcal{C}}^{\ast }\cap {\mathbb{S}}_1\left(0\right)}\left(\lambda \right)⟹\eta =0$holds.

Then, for each $y^{\ast }\in {\mathbb{R}}^m$, we have $D^{\ast }{\mathrm{\Psi }}_{\mathrm{w}}(\overline{\mathrm{x}},\overline{\mathrm{y}})\left({\mathrm{y}}^{\ast }\right)\subset \left\{{\mathrm{x}}^{\ast }\right|\begin{array}{rl}& (x^{\ast },\eta )\in D^{\ast }{\mathrm{\Psi }}_{\mathrm{w}}^{\mathrm{sc}}\left(\right(\overline{\mathrm{x}},\lambda ),\overline{\mathrm{y}})\left({\mathrm{y}}^{\ast }\right),\\ & \lambda \in \mathrm{\Xi }(\overline{x},\overline{y}),\eta \in -N_{{\mathcal{C}}^{\ast }\cap {\mathbb{S}}_1\left(0\right)}\left(\lambda \right)\end{array}\}.$

Proof.

Defining $\mathrm{{\rm Y}}:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^n\times {\mathbb{R}}^q$ by $\mathrm{{\rm Y}}\left(x\right):=\left\{x\right\}\times ({\mathcal{C}}^{\ast }\cap {\mathbb{S}}_1(0\left)\right)$ for each $x\in {\mathbb{R}}^n$, we find ${\mathrm{\Psi }}_{\mathrm{w}}={\mathrm{\Psi }}_{\mathrm{w}}^{\mathrm{sc}}\circ \mathrm{{\rm Y}}$ since Assumption 4.5 holds. Since Ξ is inner semicompact at $(\overline{x},\overline{y})$ w.r.t. its domain, the intermediate mapping $(x,y)\mapsto \left\{\right(x,\lambda )\in \mathrm{{\rm Y}}(x)|y\in {\mathrm{\Psi }}_{\mathrm{w}}^{\mathrm{sc}}(\mathrm{x},\lambda \left)\right\}=\left\{\right(\mathrm{x},\lambda )|\lambda \in \mathrm{\Xi }(\mathrm{x},\mathrm{y}\left)\right\}$is inner semicompact at $(\overline{x},\overline{y})$ w.r.t. its domain as well. Thus, the result follows from the chain rule stated in [46, Theorem 5.2] while observing that $D^{\ast }\mathrm{{\rm Y}}(\overline{x},(\overline{x},\lambda \left)\right)(x^{\ast },\eta )=\{\begin{array}{ll}\left\{x^{\ast }\right\}& \eta \in -N_{{\mathcal{C}}^{\ast }\cap {\mathbb{S}}_1\left(0\right)}\left(\lambda \right),\\ \mathrm{\varnothing }& \mathrm{otherwise}\end{array}$holds for each $\lambda \in \mathrm{\Xi }(\overline{x},\overline{y})$ and $(x^{\ast },\eta )\in {\mathbb{R}}^n\times {\mathbb{R}}^q$ while each of the additionally postulated assumptions guarantees validity of the qualification condition which is imposed in [46, Theorem 5.2].

Let us note that Lemma 4.7 generalizes [19, Proposition 3.2] to arbitrary ordering cones while coming along with a simpler proof and another type of qualification condition. Furthermore, we would like to mention that Lemma 4.7 can be used to derive necessary optimality conditions for (${\mathcal{E}}_{\mathrm{w}}$-BPP(4) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)=\phi (x),y\in \mathrm{\Gamma }(x\left)\right\}$(4) ) directly.

In order to obtain a valuable coderivative estimate for the weak frontier mapping from Lemmas 4.6 and 4.7, it remains to estimate the appearing coderivative of ${\mathrm{\Psi }}_{\mathrm{w}}^{\mathrm{sc}}$. This, however, is a well-established subject in variational analysis since ${\mathrm{\Psi }}_{\mathrm{w}}^{\mathrm{sc}}$ is the solution mapping of a scalar parametric optimization problem. Exemplary, let us mention that, in the presence of Assumption 4.5, (P$(x,\lambda )$(P(x,λ)) $\underset{y}{min}\{〈\lambda ,f(x,y)〉|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x,λ)) ) is a convex optimization problem for each $x\in {\mathbb{R}}^n$ and $\lambda \in {\mathcal{C}}^{\ast }$. Assuming, additionally, that f is a smooth function, we find $\mathrm{\forall x}\in {\mathbb{R}}^n\mathrm{\forall \lambda }\in {\mathcal{C}}^{\ast }:{\mathrm{\Psi }}_{\mathrm{w}}^{\mathrm{sc}}(\mathrm{x},\lambda )=\{\mathrm{y}|-{\mathrm{f}}_{\mathrm{y}}^{\prime }(\mathrm{x},\mathrm{y}{)}^{\mathrm{\top }}\lambda \in {\mathrm{N}}_{\mathrm{\Gamma }\left(\mathrm{x}\right)}(\mathrm{y})\},$i.e. ${\mathrm{\Psi }}_{\mathrm{w}}^{\mathrm{sc}}$ possesses the structure of a so-called parametric variational system. Consulting e.g. [38, Section 4.4], coderivative estimates for ${\mathrm{\Psi }}_{\mathrm{w}}^{\mathrm{sc}}$ can be obtained in terms of derivatives of f and coderivatives of the normal cone mapping $(x,y)\mapsto N_{\mathrm{\Gamma }\left(x\right)}\left(y\right)$ under suitable constraint qualifications. The remaining coderivative of the normal cone map may be specified for diverse different types of constraint systems, see e.g. [55–58]. Exemplary, this has been worked out in the proof of [19, Corollary 3.4] in the particular situation where $\mathcal{C}:={\mathbb{R}}_{+}^q$ and Γ is induced by smooth parametric inequality constraints.

As the purpose of this note is mainly focused on the value behind the model reformulations ($\mathcal{E}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ) and (${\mathcal{E}}_{\mathrm{w}}$-VFR(5) $\underset{x,y}{min}{}_{\mathcal{K}}\left\{F\right(x,y)|x\in X,f(x,y)\in \{\phi \left(x\right)\}-{\mathbb{R}}_{+},y\in \mathrm{\Gamma }(x\left)\right\},$(5) ), we abstain from the postulation of any more technical results, but simply combine Lemmas 4.6 and 4.7 to obtain a coderivative estimate which can be made fully explicit by respecting the above comments by the interested reader.

Theorem 4.8

Fix $(\overline{x},\overline{z})\in \mathrm{gph}{\mathrm{\Phi }}_{\mathrm{w}}$ and let $\mathrm{gph}{\mathrm{\Phi }}_{\mathrm{w}}$ be locally closed around $(\overline{x},\overline{z})$. Furthermore, let Assumption 4.5 be valid. Let ${\mathrm{\Psi }}_{\mathrm{w}}$ be inner semicompact at $\overline{x}$ w.r.t. $\mathrm{dom}{\mathrm{\Psi }}_{\mathrm{w}}$. Furthermore, for each $\overline{y}\in {\mathrm{\Psi }}_{\mathrm{w}}\left(\overline{\mathrm{x}}\right)$ with $f(\overline{x},\overline{y})=\overline{z}$, let $\mathrm{gph}{\mathrm{\Psi }}_{\mathrm{w}}$ be closed locally around $(\overline{x},\overline{y})$, let f be locally Lipschitz continuous at $(\overline{x},\overline{y})$, and, for each $\lambda \in \mathrm{\Xi }(\overline{x},\overline{y})$, let $\mathrm{gph}{\mathrm{\Psi }}_{\mathrm{w}}^{\mathrm{sc}}$ be closed locally around $\left(\right(\overline{x},\lambda ),\overline{y})$, and let one of the conditions (a) or (b) of Lemma 4.7 be valid. Then, for each $z^{\ast }\in {\mathbb{R}}^q$, $D^{\ast }{\mathrm{\Phi }}_{\mathrm{w}}(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)\subset \{{\mathrm{x}}_1^{\ast }+{\mathrm{x}}_2^{\ast }|\begin{array}{rl}& \overline{y}\in {\mathrm{\Psi }}_{\mathrm{w}}\left(\overline{\mathrm{x}}\right),\mathrm{f}(\overline{\mathrm{x}},\overline{\mathrm{y}})=\overline{\mathrm{z}},\\ & (x_1^{\ast },y^{\ast })\in D^{\ast }f(\overline{x},\overline{y})\left(z^{\ast }\right),\\ & (x_2^{\ast },\eta )\in D^{\ast }{\mathrm{\Psi }}_{\mathrm{w}}^{\mathrm{sc}}\left(\right(\overline{\mathrm{x}},\lambda ),\overline{\mathrm{y}})\left({\mathrm{y}}^{\ast }\right),\\ & \lambda \in \mathrm{\Xi }(\overline{x},\overline{y}),\eta \in -N_{{\mathcal{C}}^{\ast }\cap {\mathbb{S}}_1\left(0\right)}\left(\lambda \right)\end{array}\}.$If, additionally, f is continuously differentiable at all points $(\overline{x},\overline{y})$ such that $\overline{y}\in {\mathrm{\Psi }}_{\mathrm{w}}\left(\overline{\mathrm{x}}\right)$ and $f(\overline{x},\overline{y})=\overline{z}$ hold, then, for each $z^{\ast }\in {\mathbb{R}}^q$, $\begin{array}{rl}& D^{\ast }{\mathrm{\Phi }}_{\mathrm{w}}(\overline{\mathrm{x}},\overline{\mathrm{z}})\left({\mathrm{z}}^{\ast }\right)\\ & \subset \left\{f_x^{\prime }\right(\overline{x},\overline{y}{)}^{\mathrm{\top }}{z}^{\ast }+x^{\ast }\left|\begin{array}{rl}& \overline{y}\in {\mathrm{\Psi }}_{\mathrm{w}}\left(\overline{\mathrm{x}}\right),\mathrm{f}(\overline{\mathrm{x}},\overline{\mathrm{y}})=\overline{\mathrm{z}},\\ & (x^{\ast },\eta )\in D^{\ast }{\mathrm{\Psi }}_{\mathrm{w}}^{\mathrm{sc}}\left(\right(\overline{\mathrm{x}},\lambda ),\overline{\mathrm{y}})({\mathrm{f}}_{\mathrm{y}}^{\prime }(\overline{\mathrm{x}},\overline{\mathrm{y}}{)}^{\mathrm{\top }}{\mathrm{z}}^{\ast }),\\ & \lambda \in \mathrm{\Xi }(\overline{x},\overline{y}),\eta \in -N_{{\mathcal{C}}^{\ast }\cap {\mathbb{S}}_1\left(0\right)}\left(\lambda \right)\end{array}\right\}.\end{array}$

Let us note that, based on Lemma 3.5, one can easily check that the numerous closedness assumptions as well as the inner semicompactness assumption on ${\mathrm{\Psi }}_{\mathrm{w}}$ in Theorem 4.8 are inherently satisfied whenever Γ is lower semicontinuous w.r.t. its domain locally around $\overline{x}$, and locally bounded at $\overline{x}$. Both assumptions are inherent whenever $\mathrm{\Gamma }\left(x\right):=\mathrm{\Omega }$ holds for some nonempty and compact set $\mathrm{\Omega }\subset {\mathbb{R}}^m$ and all $x\in {\mathbb{R}}^n$. Then it remains to validate condition (a) or (b) of Lemma 4.7.

Remark 4.9

Let Assumption 4.5 be valid and ${\phi }^{\mathrm{sc}}:{\mathbb{R}}^{\mathrm{n}}\times {\mathbb{R}}^{\mathrm{q}}\to \overline{\mathbb{R}}$ be the optimal value function associated with (P$(x,\lambda )$(P(x,λ)) $\underset{y}{min}\{〈\lambda ,f(x,y)〉|y\in \mathrm{\Gamma }(x\left)\right\}.$(P(x,λ)) ), i.e. $\mathrm{\forall x}\in {\mathbb{R}}^n\mathrm{\forall \lambda }\in {\mathbb{R}}^q:{\phi }^{\mathrm{sc}}(\mathrm{x},\lambda ):=\underset{\mathrm{y}}{inf}\{〈\lambda ,\mathrm{f}(\mathrm{x},\mathrm{y})〉|\mathrm{y}\in \mathrm{\Gamma }(\mathrm{x}\left)\right\}.$Then we find the representation $\mathrm{\forall x}\in {\mathbb{R}}^n:{\mathrm{\Phi }}_{\mathrm{w}}\left(\mathrm{x}\right)=\{\mathrm{z}\in \mathrm{\Sigma }(\mathrm{x})|\mathrm{\exists \lambda }\in {\mathcal{C}}^{\ast }\cap {\mathbb{S}}_1(0):〈\lambda ,\mathrm{z}〉={\phi }^{\mathrm{sc}}(\mathrm{x},\lambda \left)\right\},$where $\mathrm{\Sigma }:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^q$ has been defined in (3(3) $\mathrm{\forall x}\in {\mathbb{R}}^n:\mathrm{\Sigma }\left(x\right):=f(x,\mathrm{\Gamma }(x\left)\right).$(3) ). Using $\mathrm{{\rm Y}}:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^n\times {\mathbb{R}}^q$ from the proof of Lemma 4.7 as well as ${\mathrm{{\rm Y}}}_1:{\mathbb{R}}^n\times {\mathbb{R}}^q\rightrightarrows {\mathbb{R}}^q\times \mathbb{R}$ and ${\mathrm{{\rm Y}}}_2:{\mathbb{R}}^q\times \mathbb{R}\rightrightarrows {\mathbb{R}}^q$ given by $\begin{array}{rl}\mathrm{\forall x}\in {\mathbb{R}}^n\mathrm{\forall \lambda }\in {\mathbb{R}}^q\mathrm{\forall \alpha }\in \mathbb{R}:{\mathrm{{\rm Y}}}_1(x,\lambda )& :=\left\{\right(\lambda ,{\phi }^{\mathrm{sc}}(\mathrm{x},\lambda )\left)\right\},\\ {\mathrm{{\rm Y}}}_2(\lambda ,\alpha )& :=\{z|〈\lambda ,z〉-\alpha =0\},\end{array}$we find ${\mathrm{\Phi }}_{\mathrm{w}}\left(\mathrm{x}\right)=\mathrm{\Sigma }\left(\mathrm{x}\right)\cap ({\mathrm{{\rm Y}}}_2\circ ({\mathrm{{\rm Y}}}_1\circ \mathrm{{\rm Y}}\left)\right)\left(\mathrm{x}\right)$. Seemingly, this representation offers the possibility to obtain an upper estimate for the coderivative of ${\mathrm{\Phi }}_{\mathrm{w}}$ via an intersection rule, see e.g. [38, Proposition 3.20], and the chain rule, see e.g. [46, Theorem 5.2]. Indeed, in the case where ${\phi }^{\mathrm{sc}}$ is a locally Lipschitzian function, which holds under fairly mild assumptions, one can apply the chain rule from [46] twice in order to find an upper estimate of the coderivative of ${\mathrm{{\rm Y}}}_2\circ ({\mathrm{{\rm Y}}}_1\circ \mathrm{{\rm Y}})$ which comprises the limiting subdifferential of (a non-specified multiple of) ${\phi }^{\mathrm{sc}}$. The latter, however, cannot be approximated from above by the Cartesian product of the associated subdifferentials w.r.t. the variables x and λ in general which is a first difficulty. A second problem pops up as the constraint qualification from [38, Proposition 3.20] which is sufficient to get the intersection rule working is also likely to fail. Hence, we abstain here from providing the details as this approach does not seem too reasonable.

Remark 4.10

In scalar bilevel optimization, the value function reformulation is often preferred over the formulation comprising the lower-level solution mapping when the derivation of necessary optimality conditions is considered. This is due to the fact that the computation of upper estimates for generalized derivatives of the lower-level value function is often much easier than obtaining upper estimates for generalized derivatives of the lower-level solution mapping. Thus, on the one hand, Lemma 4.6 and Theorem 4.8 seemingly are of limited practical use in the context of necessary optimality conditions as the coderivative of some solution mapping associated with the lower-level problem appears. Indeed, one could try to directly make use of Lemma 4.7 in order to obtain necessary optimality conditions. On the other hand, the value function approach to bilevel optimization is often preferred for the construction of solution algorithms, and taking this aspect into account, some upper estimate for the coderivative of the weak frontier mapping might be essential, and this is where Lemma 4.6 and Theorem 4.8 may become handy.

5. Conclusions

In this paper, we visualized that multiobjective bilevel optimization is, up to a certain degree, only reasonable if the lower-level efficiency-type and frontier-type mapping possess closed graphs as this is a fundamental assumption in order to guarantee the existence of solutions (in whatever sense) and to allow for the derivation of optimality conditions. We presented conditions which guarantee validity of these closedness properties, gave existence results, and discussed the computation of upper coderivative estimates for frontier-type mappings. The latter is an essential step on the route to necessary optimality conditions as visualized in the recent paper [24] which, however, has some flaws which we carved out here. Furthermore, we commented on different ways on how to introduce a value function reformulation of a multiobjective bilevel optimization problem which might be important in algorithmic applications.

In the future, it has to be investigated whether the value function reformulation of a multiobjective optimization problem can be exploited beneficially for its algorithmic solution. Whenever the lower-level problem is a scalar optimization problem, this has already been demonstrated in several different ways in the literature, see e.g. [5, Section 20.6.4] for an overview. Furthermore, it remains open whether the approach from Section 3.3 is suitable from the viewpoint of optimality conditions as, in contrast to Section 4, it is not clear how the derivation of coderivative estimates for the mappings $\overline{\mathrm{\Phi }}$ and $\overline{\mathrm{\Psi }}$ can be done.

Disclosure statement

No potential conflict of interest was reported by the author(s).