w-Invexity and optimality problem

Abstract

As the development of a family of preinvex functions, we define the w-invex set and a class of real functions. These are functions with the names w-preinvex, w-strictly preinvex, w-prequasi-invex, w-strictly prequasi-invex, w-semi-strictly prequasi-invex, and w-pre-pseudo-invex. The fundamental characteristics of these functions are thoroughly examined. Several examples are provided here that serve to clarify the concept in this context. Several of these preinvex functions are addressed to investigate the optimization problems further.

Keywords:

• Invexity
• inequality
• w-invex sets
• w-preinvex functions
• monotonicity
• optimization

1. Introduction

One analytical aspect of mathematical programming, dependent on convexity is the determination of appropriate optimality criteria. Among the various generalized convex functions, the invex functions stand out as a seminal contribution, pioneered by Hanson [1]. Over time, this concept has undergone significant refinement and extension. Notably, Zalmai [2] introduced the transformative $\rho -(\eta ,\theta )$-invex functions as a generalization of the invex functions.

The exploration of semistrictly preinvex functions on ${\mathbb{R}}^n$ was initiated by Yang and Li [3], while Rapcsak [4] gave the idea of the geodesic convexity, particularly relevant in linear spaces and its extension to Riemannian manifolds. In the realm of Riemannian spaces, Udriste [5] laid the foundation for convex programming as well as duality conditions, with Pini [6] contributing to the development of invex functions in this context. Addressing vector programming problems on differentiable manifolds, Mititelu [7] delineated necessary and sufficient conditions of Karush–Kuhn–Tucker (KKT) type, while Ferrara and Mititelu proposed the Mond-Weir form of duality for vector programming problems on such manifolds [8].

The formulation of the Fritz-John condition for Inequality Constrained Optimization Problems (ICOP) was facilitated through the utilization of alternative theorems and assumptions concerning semi-preinvex functions, as introduced by Jeyakumar [9]. Hanson, Mond, and Rueda established pivotal relationships between KKT points and mathematical programming problems [1, 10].

Recent advancements include the introduction of generalized invexity by Caristi et al. [11], the proposal of s-type preinvex functions by Ihsan et al. [12], and the delineation of a new class of preinvex functions by Noor et al. [13]. Additionally, Kiliçman et al. introduced generalized preinvex functions [14], alongside several other notable generalizations proposed by various authors [6, 7, 15–26].

In case X is not invex set w.r.t. a map η, then w.r.t. map η and another map w, we define a new set Y known as w-invex set. The motivation for defining such sets came due to the possibility of better results that were not usually obtained on invex sets. This w-invex set, a subset of ${\mathbb{R}}^n$, is a generalization of an invex set X (w.r.t. the map η). This special generalization arises in the situation when the results of optimizations are not evolved on proper invex sets, while in the case w-invex set, we get more fruitful results.

There is a connection between several tested generalizations discussed in this work and w-invexity. Generalized convexity and related ideas are introduced in Section 2. In Section 3, it is demonstrated that there is a very obvious method to follow w-prequasi-invexity, w-strictly prequasi-invexity, and w-semi-strictly prequasi-invexity for the real functions. Given that w-preinvex functions produce the necessary results, it is simple to convert these class functions into a class of regular invex functions.

2. Preliminaries

For our study, which is detailed in the following sections, we will mostly use the following definitions:

Definition 2.1

[27, 28]

A non-empty subset X of ${\mathbb{R}}^n$ is known as invex if ∃ a bi-function $\eta :{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ such that ∀$z_1,z_2\in X$ and ∀$\delta \in [0,1]$, we obtain $z_2+\mathrm{\delta \eta }(z_1,z_2)\in X$.

Definition 2.2

[27, 28]

On a non-empty subset X of ${\mathbb{R}}^n$, a function $h:X\to \mathbb{R}$ is known as preinvex if ∃$\eta :{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$, a bi-function, such that ∀$z_1,z_2\in X$ and ∀$\delta \in [0,1]$, we obtain $h(z_2+\mathrm{\delta \eta }(z_1,z_2\left)\right)\le \mathrm{\delta h}\left(z_1\right)+(1-\delta )h\left(z_2\right).$

Definition 2.3

[29]

On a non-empty subset X of ${\mathbb{R}}^n$, a function $h:X\to \mathbb{R}$ is known as prequasi invex if ∃$\eta :{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$, a bi-function, s.t., ∀$z_1,z_2\in X$ and ∀$\delta \in [0,1]$, we obtain $h(z_2+\mathrm{\delta \eta }(z_1,z_2\left)\right)\le max\left\{h\right(z_1),h(z_2\left)\right\}.$

Definition 2.4

[25]

On a non-empty subset X of ${\mathbb{R}}^n$, a function $h:X\to \mathbb{R}$ is known as strictly prequasi invex if ∃ a bi-function $\eta :{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ such that ∀$z_1,z_2\in X$, $z_1\ne z_2$ and ∀$\delta \in (0,1)$,we obtain $h(z_2+\mathrm{\delta \eta }(z_1,z_2\left)\right)<max\left\{h\right(z_1),h(z_2\left)\right\}.$

Definition 2.5

[25]

On a non-empty subset X of ${\mathbb{R}}^n$, a function $h:X\to \mathbb{R}$ is known as semi-strictly prequasi invex if ∃ a bi-function $\eta :{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ such that ∀$z_1,z_2\in X$, $h\left(z_1\right)\ne h\left(z_2\right)$ and ∀$\delta \in (0,1)$, we obtain $h(z_2+\mathrm{\delta \eta }(z_1,z_2\left)\right)<max\left\{h\right(z_1),h(z_2\left)\right\}.$

The following definitions and remarks are introduced in this paper

Definition 2.6

A non-empty subset X of ${\mathbb{R}}^n$ is said to be w-invex if there exists map $\eta :{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ and $w:{\mathbb{R}}^n\to {\mathbb{R}}^n$ such that $\begin{array}{rl}& w\left(z_2\right)+\mathrm{\delta \eta }(z_1,w(z_2\left)\right)\in X,\mathrm{\forall }{z}_1,z_2\in X\text{and}\\ & \delta \in [0,1].\end{array}$

Remark 2.1

Every invex set is w-invex (as take w is an identity map) however converse isn't always true. See example (3.1).

Definition 2.7

Let X be a w-invex set. A function $h:X\to \mathbb{R}$ is said to be

1. w-preinvex if ∀ points $z_1,z_2\in X$, and $0\le \delta \le 1$, such that $h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)\le \mathrm{\delta h}\left(z_1\right)+(1-\delta )h\left(z_2\right).$

2. w-strictly preinvex if ∀ points $z_1,z_2\in X$, $z_1\ne z_2$ and $0<\delta <1$, such that $h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)<\mathrm{\delta h}\left(z_1\right)+(1-\delta )h\left(z_2\right).$

3. w-prequasi-invex if ∀ points $z_1,z_2\in X$, and $0\le \delta \le 1$, such that $h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)\le \text{max}\left\{h\right(z_1),h(z_2\left)\right\}.$

4. w-strictly prequasi-invex if ∀ points $z_1,z_2\in X$, $z_1\ne z_2$ and $0<\delta <1$, such that $h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)<\text{max}\left\{h\right(z_1),h(z_2\left)\right\}.$

5. w-semi-strictly prequasi-invex if ∀ points $z_1,z_2\in X$, $h\left(z_1\right)\ne h\left(z_2\right)$ and $0<\delta <1$, such that $h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)<\text{max}\left\{h\right(z_1),h(z_2\left)\right\}.$

Remark 2.2

In general, every w-strictly preinvex function is also a w-preinvex function; every w-preinvex function is also a w-prequasi-invex function; every preinvex function is also a w-preinvex function (as take w is an identity map); however, converse isn't always true. See examples (3.4) and (3.5).

3. w-invexity

We construct the following examples to verify/support the Definition (2.6) onwards

Example 3.1

Let $\eta :[0,\mathrm{\infty })\times [0,\mathrm{\infty })\to \mathbb{R}$ be given by: $\eta (z_1,z_2)=z_1(z_2-2),$and let the function $w:[0,\mathrm{\infty })\to \mathbb{R}$ be given by: $w\left(z_2\right)=z_2+2.$Then, $[0,\mathrm{\infty })$ is not an invex set w.r.t. (w.r.t. referred to with respect to) map η but it is w-invex set w.r.t. the maps η and w.

Example 3.2

Consider $h:\mathbb{R}\to \mathbb{R},$ be defined as: $h\left(z_1\right)=z_1+k,$ where k is a constant.

Let the function $w:\mathbb{R}\to \mathbb{R}$ be defined as: $w\left(z_2\right)=z_2-7,$ and consider $\eta :\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ as: $\eta (z_1,z_2)=z_1-z_2-6$ for all $z_1,z_2\in \mathbb{R}$. Then, the function h is w-preinvex function w.r.t. η and w.

Example 3.3

Consider $h:\mathbb{R}\to \mathbb{R},$ be defined as: $h\left(z_1\right)=z_1+k,$ where k is a constant.

Consider the function $w:\mathbb{R}\to \mathbb{R}$ be defined as: $w\left(z_2\right)=z_2,$ and consider $\eta :\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ be defined as: $\eta (z_1,z_2)=z_1-z_2$ for all $z_1,z_2\in \mathbb{R}$. Then, the function h is w-preinvex function w.r.t. η and w, but not w-strictly preinvex and preinvex function.

Example 3.4

Consider $h:\mathbb{R}\to \mathbb{R},$ be defined as: $h\left(z_1\right)=z_1^5$. Consider the function $w:\mathbb{R}\to \mathbb{R}$ be defined as: $w\left(z_2\right)=z_2-6,$ and consider $\eta :\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ be defined as: $\eta (z_1,z_2)=z_1-z_2-6$ for all $z_1,z_2\in \mathbb{R}$. Then, the function h is w-prequasi-invex function w.r.t. η and w, but not w-preinvex function.

Example 3.5

Let $h:[0,\mathrm{\infty })\to \mathbb{R},$ be defined as: $h\left(z_1\right)=z_1+k,$ where k is a constant. let the function $w:[0,\mathrm{\infty })\to \mathbb{R}$ be defined as: $w\left(z_2\right)=0,$ and consider $\eta :[0,\mathrm{\infty })\times [0,\mathrm{\infty })\to \mathbb{R}$ be defined as: $\eta (z_1,z_2)=z_1+2z_2$ for all $z_1,z_2\in [0,\mathrm{\infty })$. It is simple to demonstrate that while h is w-preinvex w.r.t. η and w and, it is not preinvex w.r.t. η.

Example 3.6

Let $h:[0,\mathrm{\infty })\to \mathbb{R},$ be defined as: $h\left(z_1\right)=\{\begin{array}{l}11,\mathrm{if}{z}_1\in [0,11)\\ -11,\mathrm{if}{z}_1\in [11,\mathrm{\infty }),\end{array}$let the function $w:[0,\mathrm{\infty })\to \mathbb{R}$ be defined as: $w\left(z_2\right)=z_2+11,$ and consider $\eta :[0,\mathrm{\infty })\times [0,\mathrm{\infty })\to \mathbb{R}$ be defined as: $\eta (z_1,z_2)=z_1^2+z_2^2+11$ for all $z_1,z_2\in [0,\mathrm{\infty })$. Then, the function h is w-prequasi-invex function w.r.t. η and w.

Now, consider the following main result

Theorem 3.1

Let X be a non-empty w-invex subset of ${\mathbb{R}}^n$. $h:X\to \mathbb{R}$ is w-preinvex function w.r.t. η and w if and only if the epigraph of h, $\mathrm{epi}\left(h\right)=\left\{\right(z,\alpha ):\alpha \in \mathbb{R}\mathrm{and}h(z)\le \alpha \},$is w-invex w.r.t. the same η and w.

Proof.

Consider $h:X\to \mathbb{R}$ be a w-preinvex function w.r.t. η and w. Suppose that $(z_1,\alpha )$ and $(z_2,\text{β})\in \mathrm{epi}\left(h\right)$, with $z_1,z_2\in X$, $h\left(z_1\right)\le \alpha$ and $h\left(z_2\right)\le \text{β}$, for $\alpha ,\text{β}\in \mathbb{R}$. Thus, $\begin{array}{rl}& h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)\\ & \le \mathrm{\delta h}\left(z_1\right)+(1-\delta )h\left(z_2\right).\\ & =\mathrm{\delta \alpha }+(1-\delta )\text{β},\text{for}\delta \in [0,1]\end{array}$which implies that $\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)),\mathrm{\delta \alpha }+(1-\delta \left)\text{β}\right)\in \mathrm{epi}\left(h\right).$This shows that $\mathrm{epi}\left(h\right)$ is w-invex w.r.t. the maps η and w. Conversely, assume that $\mathrm{epi}\left(h\right)$ is w-invex set w.r.t. the maps η and w. Take $z_1,z_2\in X$. Then $(z_1,h(z_1\left)\right),(z_2,h(z_2\left)\right)\in \mathrm{epi}\left(h\right)$, it follows that $\begin{array}{rl}& \left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)),\mathrm{\delta h}(z_1)\\ & +(1-\delta )h\left(z_2\right))\in \mathrm{epi}(h),\mathrm{\forall }\delta \in [0,1],\end{array}$then $h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)\le \mathrm{\delta h}\left(z_1\right)+(1-\delta )h\left(z_2\right).$By doing so, it is demonstrated that h is a w-preinvex function w.r.t. the map η and w.

Theorem 3.2

Let $h:X\to \mathbb{R}$ be a w-preinvex function w.r.t. the functions η and w, where X is any non-empty w-invex subset of ${\mathbb{R}}^n$. Then, the level set $M_{\alpha }=\{z\in X,h(z)\le \alpha \mathrm{and}\alpha \in \mathbb{R}\}$is w-invex w.r.t. the same maps η and w.

Proof.

Let $z_1,z_2\in M_{\alpha }$. Then $z_1,z_2\in X$, $h\left(z_1\right)\le \alpha \text{and}h\left(z_2\right)\le \alpha$. Since h is w-preinvex then, we have, $\mathrm{\forall \delta }\in [0,1]$, $\begin{array}{rl}& h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)\\ & \le \mathrm{\delta h}\left(z_1\right)+(1-\delta )h\left(z_2\right).\\ & =\mathrm{\delta \alpha }+(1-\delta )\alpha ,\text{for}\delta \in [0,1]\\ & =\alpha .\end{array}$This shows that $w\left(z_2\right)+\mathrm{\delta \eta }(z_1,w(z_2\left)\right)\in X$, and the level set $M_{\alpha }$ is a w-invex set w.r.t. the map η and w.

Theorem 3.3

Let X be a non-empty w-invex subset of ${\mathbb{R}}^n$. The function $h:X\to \mathbb{R}$ is w-prequasi-invex w.r.t. the same η and w if and only if the level set $M_{\alpha }=\{z\in X,h(z)\le \alpha \text{and}\alpha \in \mathbb{R}\}$is w-invex w.r.t. the same η and w.

Proof.

Suppose that the function h is w-prequasi-invex w.r.t. the maps η and w. Let $z_1,z_2\in M_{\alpha }.$ Then, $z_1,z_2\in X$ and $\text{max}\left\{h\right(z_1),h(z_2\left)\right\}\le \alpha$. Since X is a w-invex set, so, $\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right))\in X,\mathrm{\forall }\delta \in [0,1].$Since h is w-prequasi-invex function, we have $h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)\le \text{max}\left\{h\right(z_1),h(z_2\left)\right\}\le \alpha ,$so $\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right))\in M_{\alpha }.$This implies that $M_{\alpha }$ is a w-invex set w.r.t. the maps η and w.

Conversely, assume that the level set $M_{\alpha }$ is w-invex w.r.t. the maps η and w. Let $z_1,z_2\in M_{\alpha }$ for $\text{max}\left\{h\right(z_1),h(z_2\left)\right\}=\alpha .$ Since X is a w-invex set, we have $w\left(z_2\right)+\mathrm{\delta \eta }(z_1,w(z_2\left)\right)\in X,$implies that $h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)\le \text{max}\left\{h\right(z_1),h(z_2\left)\right\}=\alpha .$This prove that h is a w-prequasi-invex function w.r.t. the functions η and w.

Theorem 3.4

Assume that $h:X\to \mathbb{R}$ is a w-preinvex function w.r.t. the maps η and w, where X is any non-empty w-invex subset of ${\mathbb{R}}^n$ and $\nu =\underset{z\in X}{inf}h\left(z\right).$ Then, the set $F=\{z\in X:h(z)=\nu \}$is w-invex w.r.t. the same η and w. If h is w-strictly preinvex then F is a singleton.

Proof.

Assume that $z_1,z_2\in X$ and $\delta \in [0,1]$. Since h is a w-preinvex then, we have $\begin{array}{rl}& h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)\\ & \le \mathrm{\delta h}\left(z_1\right)+(1-\delta )h\left(z_2\right)\\ & =\mathrm{\delta \nu }+(1-\delta )\nu ,\text{for}\delta \in [0,1]\\ & =\nu .\end{array}$This follows that $w\left(z_2\right)+\mathrm{\delta \eta }(z_1,w(z_2\left)\right)\in F,$which implies that F is w-invex w.r.t. the maps η and w.

For the other part, contrarily suppose that $h\left(z_1\right)=h\left(z_2\right)=\nu$. Since X is a w-invex set, so $w\left(z_2\right)+\mathrm{\delta \eta }(z_1,w(z_2\left)\right)\in F,\mathrm{\forall }\delta \in [0,1].$Also, since h is w-strictly preinvex, we have $\begin{array}{rl}& h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)\\ & <\mathrm{\delta h}\left(z_1\right)+(1-\delta )h\left(z_2\right)\\ & =\mathrm{\delta \nu }+(1-\delta )\nu ,\text{for}\delta \in [0,1]\\ & =\nu .\end{array}$This contradiction the fact that $\nu =\underset{z\in X}{inf}h\left(z\right).$ Therefore, F is singleton.

Theorem 3.5

Let $h:X\to \mathbb{R}$ is a w-preinvex function on X, where X is any non-empty w-invex subset of ${\mathbb{R}}^n$. Therefore, any local minimum point is also a global minimum point.

Proof.

Let $z_2$ be the local minimum point for h. Then, a neighbourhood N of $z_2$ such that (1) $h\left(z_1\right)\ge h\left(z_2\right),\mathrm{\forall }{z}_1\in N\cap X.$(1) Assuming that $z_2\in X$ is not a global minimum point of h, then ∃ a point $z\in X$ $h\left(z\right)<h\left(z_2\right).$Since h is w-preinvex X, ∃η and w such that, ∀$\delta \in [0,1]$, we have (2) $\begin{array}{rl}h\left(w\right(z_2)+\mathrm{\delta \eta }(z,w\left(z_2\right)\left)\right)& \le \mathrm{\delta h}\left(z\right)+(1-\delta )h\left(z_2\right)\\ & <\mathrm{\delta h}\left(z_2\right)+(1-\delta )h\left(z_2\right)\\ & =h\left(z_2\right).\end{array}$(2) As $w\left(z_2\right)+\mathrm{\delta \eta }(z_1,w(z_2\left)\right)\in X$ for each $\delta \in [0,1]$, ∃$ϵ>0$ such that $w\left(z_2\right)+\mathrm{\delta \eta }(z_1,w(z_2\left)\right)\in X\cap N$ for each $\delta \in [0,ϵ)$, in a contradiction with (1(1) $h\left(z_1\right)\ge h\left(z_2\right),\mathrm{\forall }{z}_1\in N\cap X.$(1) ).

Theorem 3.6

Let $h:X\to \mathbb{R}$ is a w-preinvex function on X, where X is any non-empty w-invex subset of ${\mathbb{R}}^n$. Any strict local minimum point is thus a strict global minimum point.

Proof.

The proof can be achieved on similar lines as of Theorem 3.5.

Theorem 3.7

If $h:X\to \mathbb{R}$ is w-preinvex on the w-invex set X, where X is any non-empty w-invex subset of ${\mathbb{R}}^n$, then also kh is w-preinvex w.r.t. η and w, for any k>0.

Proof.

Since h is a w-preinvex function then, we have ∀$z_1,z_2\in X$ and $\delta \in [0,1]$ $h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)\le \mathrm{\delta h}\left(z_1\right)+(1-\delta )h\left(z_2\right),$we have, for any k>0, $\begin{array}{rl}& k\left(h\right(w\left(z_2\right)+\mathrm{\delta \eta }(z_1,w(z_2\left)\right)\left)\right)\\ & \le k\left(\mathrm{\delta h}\right(z_1)+(1-\delta \left)h\right(z_2\left)\right),\mathrm{\forall }\delta \in [0,1]\end{array}$from which $\begin{array}{rl}& k\left(h\right(w\left(z_2\right)+\mathrm{\delta \eta }(z_1,w(z_2\left)\right)\left)\right)\\ & \le \delta \left(\mathrm{kh}\right)\left(z_1\right)+(1-\delta )\left(\mathrm{kh}\right)\left(z_2\right),\mathrm{\forall }\delta \in [0,1].\text{■}\end{array}$

Theorem 3.8

Let X be a non-empty w-invex subset of ${\mathbb{R}}^n$ and $h_i:X\to \mathbb{R}$, $i=1,2,\dots ,m,$ be w-preinvex w.r.t. same η and w. Then, $\sum _{i=1}^m{k}_i{h}_i\left(z\right)$ is w-preinvex w.r.t. same η and w, where $k_i>0$, $i=1,2,\dots ,m.$

Proof.

By hypothesis we have $h_i\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)\le \delta h_i\left(z_1\right)+(1-\delta )h_i\left(z_2\right).$It follows that $\begin{array}{rl}& k_i{h}_i\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)\\ & \le \delta k_i{h}_i\left(z_1\right)+(1-\delta )k_i{h}_i\left(z_2\right).\end{array}$And $\begin{array}{rl}& \left(\sum _{i=1}^m{k}_i{h}_i\right)\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)\\ & \le \delta \left(\sum _{i=1}^m{k}_i{h}_i\right)\left(z_1\right)+(1-\delta )\left(\sum _{i=1}^m{k}_i{h}_i\right)\left(z_2\right).\end{array}$Thus, we achieve the proof.

Theorem 3.9

Let $h:X\to \mathbb{R}$ be a w-preinvex w.r.t. η and w, where X is any non-empty w-invex subset of ${\mathbb{R}}^n$, let $\varphi :\mathbb{R}\to \mathbb{R}$ be an increasing and convex. Then $\varphi \circ h$ is w-preinvex w.r.t. η and w.

Proof.

As h is w-preinvex w.r.t. η and w, we obtain ∀$z_1,z_2\in X,\delta \in [0,1]$, $h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)\le \mathrm{\delta h}\left(z_1\right)+(1-\delta )h\left(z_2\right).$Due to the fact that $\varphi :\mathbb{R}\to \mathbb{R}$ non-decreasing and convex, we obtain $\begin{array}{rl}& \varphi \left(h\right(w\left(z_2\right)+\mathrm{\delta \eta }(z_1,w(z_2\left)\right)\left)\right)\\ & \le \varphi \left(\delta \right(h\left(z_1\right)+(1-\delta )h\left(z_2\right))\\ & \le \mathrm{\delta \varphi }\left(h\right(z_1\left)\right)+(1-\delta )\varphi \left(h\right(z_1\left)\right),\end{array}$or $\begin{array}{rl}& \varphi \circ h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right))\\ & \le \mathrm{\delta \varphi }\circ h\left(z_1\right)+(1-\delta )\varphi \circ h\left(z_1\right).\text{■}\end{array}$

Definition 3.1

Let X is any non-empty w-invex subset of ${\mathbb{R}}^n$. $h:X\to \mathbb{R}$ is said to be w-pre pseudo-invex function if ∃ functions η, w and a function b, positive on domain, such that $\begin{array}{rl}h\left(z_1\right)& <h\left(z_2\right)⟹h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)\\ & \le h\left(z_2\right)+\delta (\delta -1)b(z_1,z_2)\end{array}$for every $\delta \in (0,1)$ and $z_1,z_2\in X$.

Theorem 3.10

Let X be a non-empty w-invex subset of ${\mathbb{R}}^n$. If h is w-preinvex, then h is w-pre pseudo-invex w.r.t. the same η and w.

Proof.

For $h\left(z_1\right)<h\left(z_2\right)$, for every $\delta \in (0,1),$ we can write $\begin{array}{rl}& \left(h\right(w\left(z_2\right)+\mathrm{\delta \eta }(z_1,w(z_2\left)\right)\left)\right)\\ & \le h\left(z_2\right)+\delta \left[h\right(z_1)-h(z_2\left)\right]\\ & <h\left(z_2\right)+\delta \left[h\right(z_1)-h(z_2\left)\right]-{\delta }^2\left[h\right(z_1)-h(z_2\left)\right],\\ & =h\left(z_2\right)+\delta (\delta -1)\left[h\right(z_2)-h(z_1\left)\right],\end{array}$where $b(z_1,z_2)=h\left(z_2\right)-h\left(z_1\right)>0.$

Theorem 3.11

Let $h:X\to \mathbb{R}$ is a w-pre pseudo-invex function, where X is any non-empty w-invex subset of ${\mathbb{R}}^n$. Consequently, any local minimum point is also a global minimum point.

Proof.

Let us assume irrationally that $z_2$ be a local minimum point rather than a global one. Then ∃$z_1\in X$ such that $h\left(z_1\right)<h\left(z_2\right).$ According to the w-pre pseudo-invexity definition, which means that, we have ∃$b(z_1,z_2)>0$, such that, ∀$\delta \in (0,1)$, $\begin{array}{rl}& h(z_2+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)\\ & \le h\left(z_2\right)+\delta (\delta -1)b(z_1,z_2)<h\left(z_2\right),\end{array}$from which $h\left(z_2\right)>h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)$,  ∀$\delta \in (0,1),$ is contradiction with the assumption.

Theorem 3.12

Assume that $\eta (z_1,w(z_2\left)\right)\ne 0$ whenever $z_1\ne w\left(z_2\right)$. Let h be a w-prequasi-invex function on X w.r.t. η and w, where X is any non-empty w-invex subset of ${\mathbb{R}}^n$. Consequently, any strict local minimum point is h also a strict global minimum point.

Proof.

Contrarily suppose that $z_2$ be strict local minimum but not global; then ∃$z\in X$ s.t., $h\left(z_2\right)<h\left(z\right).$ Since h is w-pre quasi-invex, we have $h(z_2+\mathrm{\delta \eta }(z,w\left(z_2\right)\left)\right)\le h\left(z\right),$a contradiction.

Theorem 3.13

Let $h:X\to \mathbb{R}$ be a w-prequasi-invex function w.r.t. η, where X is any non-empty w-invex subset of ${\mathbb{R}}^n$ and w. Suppose that $\varphi :\mathbb{R}\to \mathbb{R}$ is a increasing function, then, $\varphi \circ h$ is w-prequasi-invex w.r.t. η and w.

Proof.

Given that h is w-prequasi-invex function and ϕ is a increasing, we obtain $\begin{array}{rl}& \varphi \circ h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)\\ & \le \varphi \left(\text{max}\right\{h\left(z_1\right),h\left(z_2\right)\left\}\right).\\ & \varphi \circ h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)\\ & \le \varphi \left(\text{max}\right\{h\left(z_1\right),h\left(z_2\right)\left\}\right)\\ & =\text{max}\{\varphi \circ h(z_1),\varphi \circ h(z_2\left)\right\},\end{array}$which expresses the fact that the composition function $\varphi \circ h$ is w-prequasi-invex.

4. Optimality

Let $h:{\mathbb{R}}^n\to \mathbb{R}$ and $g_i:{\mathbb{R}}^n\to \mathbb{R}$, $i=1,2,\dots ,m$ are w-preinvex functions on ${\mathbb{R}}^n$. A w-preinvex programming problem is formulated as follows: (3) $\begin{array}{rl}& minh\left(z\right)\\ & \mathrm{subjectto}\\ & \text{X}=\{z\in {\mathbb{R}}^n:g_i(z)\le 0,i=1,\dots ,m\}.\end{array}$(3)

Theorem 4.1

If $h:{\mathbb{R}}^n\to \mathbb{R}$ is w-strictly preinvex function on X, where X is any non-empty w-invex subset of ${\mathbb{R}}^n$, then the global optimal solution of problem (3(3) $\begin{array}{rl}& minh\left(z\right)\\ & \mathrm{subjectto}\\ & \text{X}=\{z\in {\mathbb{R}}^n:g_i(z)\le 0,i=1,\dots ,m\}.\end{array}$(3) ) is unique.

Proof.

Let $z_1,z_2\in X$ be different global optimal solutions of problem (3(3) $\begin{array}{rl}& minh\left(z\right)\\ & \mathrm{subjectto}\\ & \text{X}=\{z\in {\mathbb{R}}^n:g_i(z)\le 0,i=1,\dots ,m\}.\end{array}$(3) ). Then $h\left(z_1\right)=h\left(z_2\right)$. Since h is w-strictly invex then, we have $\begin{array}{rl}h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)& <\mathrm{\delta h}\left(z_1\right)+(1-\delta )h\left(z_2\right)\\ & =h\left(z_1\right),\\ h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)& <h\left(z_1\right),\text{for each}\delta \in (0,1).\end{array}$This contradicts the optimality of $z_2$ for problem (3(3) $\begin{array}{rl}& minh\left(z\right)\\ & \mathrm{subjectto}\\ & \text{X}=\{z\in {\mathbb{R}}^n:g_i(z)\le 0,i=1,\dots ,m\}.\end{array}$(3) ). Hence, the global optimal solution of problem (3(3) $\begin{array}{rl}& minh\left(z\right)\\ & \mathrm{subjectto}\\ & \text{X}=\{z\in {\mathbb{R}}^n:g_i(z)\le 0,i=1,\dots ,m\}.\end{array}$(3) ) is unique.

Theorem 4.2

Let $h:{\mathbb{R}}^n\to \mathbb{R}$ be w-prequasi-invex function on X, where X is any non-empty w-invex subset of ${\mathbb{R}}^n$ and let $\gamma =\underset{z\in X}{min}h\left(z\right)$. The set $Y=\{z\in X:h(z)=\gamma \}$ of optimal solution of problem (3(3) $\begin{array}{rl}& minh\left(z\right)\\ & \mathrm{subjectto}\\ & \text{X}=\{z\in {\mathbb{R}}^n:g_i(z)\le 0,i=1,\dots ,m\}.\end{array}$(3) ) is w-invex set. If h is w-strictly prequasi-invex, then the set Y is a singleton.

Proof.

Let $z_1,z_2\in X$ be different global optimal solutions of problem (3(3) $\begin{array}{rl}& minh\left(z\right)\\ & \mathrm{subjectto}\\ & \text{X}=\{z\in {\mathbb{R}}^n:g_i(z)\le 0,i=1,\dots ,m\}.\end{array}$(3) ). Then $h\left(z_1\right)=h\left(z_2\right)=\gamma .$Since $h:{\mathbb{R}}^n\to \mathbb{R}$ be w-prequasi-invex function on X, then $h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)\le \text{max}\left\{h\right(z_1),h(z_2\left)\right\}=\gamma$which implies that $h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right))\in Y,$ so Y is w-invex set.

For the other part, on contrary let $z_1,z_2\in Y$, $z_1\ne z_2,\delta \in (0,1),$ then $h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right))\in X.$

Further, since h is w-strictly prequasi invexity function on X, we have $h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)<\text{max}\left\{h\right(z_1),h(z_2\left)\right\}=\gamma .$This contradicts that $\gamma =\underset{z\in X}{min}h\left(z\right)$ and hence the results follows.

Theorem 4.3

Consider the function $h:{\mathbb{R}}^n\to \mathbb{R}$ is w-prequasi-invex on a w-invex subset of ${\mathbb{R}}^n$. Suppose $\gamma =\underset{z\in X}{min}h\left(z\right)$. Then, the optimal solutions of problem (3(3) $\begin{array}{rl}& minh\left(z\right)\\ & \mathrm{subjectto}\\ & \text{X}=\{z\in {\mathbb{R}}^n:g_i(z)\le 0,i=1,\dots ,m\}.\end{array}$(3) ) i.e. $Y=\{z\in X:h(z)=\gamma \}$ is a w-invex set. If h is w-strictly prequasi-invex on a w-invex subset of ${\mathbb{R}}^n$. Then w-invex set a singleton set.

Proof.

Let $z_1,z_2\in Y$ and $\delta \in [0,1]$, then $z_1,z_2\in X$ also $h\left(z_1\right)=\gamma$, $h\left(z_2\right)=\gamma$. As h is w-prequasi-invex function on w-invex set, we have $w\left(z_2\right)+\mathrm{\delta \eta }(z_2,w(z_1\left)\right)\in X$and $h\left(w\right(z_2)+\mathrm{\delta \eta }(z_2,w\left(z_1\right)\left)\right)\le max\left\{h\right(z_1),h(z_2\left)\right\}=\gamma$which shows that $\left(w\right(z_2)+\mathrm{\delta \eta }(z_2,w\left(z_1\right))\in Y,$it demonstrate that w-invex set.

For the other part, suppose to the contrary that $z_1,z_2\in X$, and $z_1\ne z_2$, $\delta \in (0,1)$, we get $w\left(z_2\right)+\mathrm{\delta \eta }(z_2,w(z_1\left)\right)\in X.$As h is w-strictly prequasi-invex on X, we get $h\left(w\right(z_2)+\mathrm{\delta \eta }(z_2,w\left(z_1\right)\left)\right)<max\left\{h\right(z_1),h(z_2\left)\right\}=\gamma ,$there is a contradiction.

Theorem 4.4

Let X be a non-empty w-invex subset of ${\mathbb{R}}^n$, and let $h:{\mathbb{R}}^n\to \mathbb{R}$ be w-strictly prequasi-invex function on X. Also, the inequality $h\left(w\right(z_2)+\mathrm{\delta \eta }(z_2,w\left(z_1\right)\left)\right)>h\left(z_2\right)+2\delta \left[h\right(z_1)-h(z_2\left)\right]$ holds for all $z_2\in X$, then the problem (3(3) $\begin{array}{rl}& minh\left(z\right)\\ & \mathrm{subjectto}\\ & \text{X}=\{z\in {\mathbb{R}}^n:g_i(z)\le 0,i=1,\dots ,m\}.\end{array}$(3) ) has an optimal solution.

Proof.

As h is w-preinvex w.r.t. η and w, we obtain ∀$z_1,z_2\in X,z_1\ne z_2,\delta \in (0,1)$, (4) $h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)<\mathrm{\delta h}\left(z_1\right)+(1-\delta )h\left(z_2\right).$(4) Also, given that (5) $\begin{array}{rl}& h\left(w\right(z_2)+\mathrm{\delta \eta }(z_2,w\left(z_1\right)\left)\right)\\ & >h\left(z_2\right)+2\delta \left[h\right(z_1)-h(z_2\left)\right],\text{for all}{z}_2\in X.\end{array}$(5) Combining the inequalities (4(4) $h\left(w\right(z_2)+\mathrm{\delta \eta }(z_1,w\left(z_2\right)\left)\right)<\mathrm{\delta h}\left(z_1\right)+(1-\delta )h\left(z_2\right).$(4) ) and (5(5) $\begin{array}{rl}& h\left(w\right(z_2)+\mathrm{\delta \eta }(z_2,w\left(z_1\right)\left)\right)\\ & >h\left(z_2\right)+2\delta \left[h\right(z_1)-h(z_2\left)\right],\text{for all}{z}_2\in X.\end{array}$(5) ), we have $\delta \left(h\right(z_1)-h(z_2\left)\right)<0,\text{for all}{z}_2\in X$so $h\left(z_1\right)-h\left(z_2\right)<0,\text{for all}{z}_2\in X.$Hence, the problem (3(3) $\begin{array}{rl}& minh\left(z\right)\\ & \mathrm{subjectto}\\ & \text{X}=\{z\in {\mathbb{R}}^n:g_i(z)\le 0,i=1,\dots ,m\}.\end{array}$(3) ) has an optimal solution of h.

Theorem 4.5

Let X be a non-empty w-invex subset of ${\mathbb{R}}^n$ w.r.t. η and w, and let $h:X\to \mathbb{R}$ be w-semi-strictly prequasi-invex function for the same η and w. If $\overline{z}\in X$ is a local optimal solution to the problem of minimizing $h\left(z\right)$ subject to $z\in X$, then $\overline{z}$ is a global minimum.

Proof.

Suppose that $\overline{z}\in X$ is a local minimum. Then, there is an ϵ-neighbourhood $N_{ϵ}\left(\overline{z}\right)$ around $\overline{z}$ such that (6) $h\left(\overline{z}\right)\le h\left(z\right),\mathrm{\forall }z\in X\cap N_{ϵ}\left(\overline{z}\right).$(6) If $\overline{z}$ is not a global minimum of h, then there exists an $z^{\ast }\in X$ such that (7) $h\left(z^{\ast }\right)<h\left(\overline{z}\right).$(7) By the w-semi-strictly preinvexity of h, and using (7(7) $h\left(z^{\ast }\right)<h\left(\overline{z}\right).$(7) ), we have $\begin{array}{rl}h\left(w\right(\overline{z})+\mathrm{\delta \eta }(z^{\ast },w\left(\overline{z}\right)\left)\right)& <\mathrm{\delta h}\left(z^{\ast }\right)+(1-\delta )h\left(\overline{z}\right)\\ & <\mathrm{\delta h}\left(\overline{z}\right)+(1-\delta )h\left(\overline{z}\right)\\ & =h\left(\overline{z}\right),\end{array}$for all $0<\delta <1.$ For a sufficiently small $\delta >0$, it follows that $w\left(\overline{z}\right)+\mathrm{\delta \eta }(z^{\ast },w(\overline{z}\left)\right)\in X\cap N_{ϵ}\left(\overline{z}\right),$which is a contradiction to (6(6) $h\left(\overline{z}\right)\le h\left(z\right),\mathrm{\forall }z\in X\cap N_{ϵ}\left(\overline{z}\right).$(6) ). This completes the proof.

Example 4.1

Suppose a company manufactures a product using labour and raw materials. The company aims to minimize its total product cost while adhering to constraints on labour and raw material availability.

The cost function for producing product A is as follows: The cost function for producing product A is given by $h(L,R)=L^2+R^2,$ where h taken as a w-preinvex function. The company faces the following constraints on labour and raw materials. Maximum of 150 units of the sum of raw and labour available $g(L,R)=L+R-150\le 0,R,L\ge 0$.

5. Conclusion

In this paper, we have introduced the w-invex set, w-preinvex, w-prequasi-invex, and w-semi-strictly prequasi-invex functions, which are generalization of the invex set, preinvex, prequasi-invex, and semi-strictly prequasi-invex functions, respectively. We give some interesting results in this context. Also, we developed some fruitful examples to illustrate these concepts. In the last, the application of these functions in solving an optimization problem is also given in the paper.

Disclosure statement

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