New representations of epigraphs of conjugate mappings and Lagrange, Fenchel–Lagrange duality for vector optimization problems

Abstract

In this paper, we concern the vector problem: $\begin{array}{r}\left(\mathrm{V}\mathrm{P}\right)\mathrm{W}\mathrm{I}\mathrm{n}\mathrm{f}\left\{F\right(x):x\in C,G(x)\in -S\},\end{array}$where X, Y, Z are locally convex Hausdorff topological vector spaces, $F:X\to Y\cup \{+{\mathrm{\infty }}_Y\}$ and $G:X\to Z\cup \{+{\mathrm{\infty }}_Z\}$ are proper mappings, C is a nonempty convex subset of X, and S is a non-empty closed, convex cone in Z. Several new presentations of epigraphs of composite conjugate mappings associated to (VP) are established under variant qualifying conditions. The significance of these representations is twofold: Firstly, they play a key role in establishing new kinds of vector Farkas lemmas which serve as tools in the study of vector optimization problems; secondly, they pay the way to define Lagrange and two new kinds of Fenchel–Lagrange dual problems for (VP). Strong and stable strong duality results corresponding to these mentioned dual problems of (VP) are established using the new Farkas-type results just obtained. It is shown that in the special case where $Y=\mathbb{R}$, the Lagrange and Fenchel–Lagrange dual problems for (VP), go back to Lagrange, and Fenchel–Lagrange dual problems for scalar problems, and the resulting duality results cover, and in some cases, extend the corresponding ones for scalar problems in the literature.

Keywords:

• Vector inequalities
• (stable) vector Farkas lemmas
• qualification conditions
• extended epigraphs of conjugate mappings
• Lagrange and Fenchel–Lagrange duality for vector optimization problems

AMS Classifications:

• 49N15
• 90C25
• 90C29
• 90C31
• 90C46
• 90C48

Disclosure statement

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

Notes

1 The problem considered in Ref. [4] is the vector composite problem which is more general than (VP). The dual problems considered there, however, are all of the Lagrange ones.

2 Here, by the term ‘decomposition’ we mean the sets in the right-hand side of the equality are disjoint.

3 This notion was used in Refs. [14,30] as ‘star K-lower semicontinuous’.

4 For other orderings on ${\mathcal{P}}_0\left(Y^{\bullet }\right)$, see, e.g. [33].

5 Observe that when $\mathrm{W}\mathrm{I}\mathrm{n}\mathrm{f}\left(\mathrm{V}\mathrm{P}\right)=\mathrm{W}\mathrm{M}\mathrm{a}\mathrm{x}\left({\mathrm{V}\mathrm{D}}_1\right)$, one has $\mathrm{W}\mathrm{S}\mathrm{u}\mathrm{p}\left({\mathrm{V}\mathrm{D}}_1\right)=\mathrm{W}\mathrm{M}\mathrm{a}\mathrm{x}\left({\mathrm{V}\mathrm{D}}_1\right)$ (see Proposition 3.2 (i)), and hence, $\left({\mathrm{V}\mathrm{D}}_1\right)$ attains at any value from $\mathrm{W}\mathrm{S}\mathrm{u}\mathrm{p}\left({\mathrm{V}\mathrm{D}}_1\right)$.

Additional information

Funding

This research was supported by the Vietnam National University - Ho Chi Minh city, Ho Chi Minh city, Vietnam (VNU- HCM) [grant number B2021-28-03].