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Research Article

w-Invexity and optimality problem

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Article: 2359207 | Received 20 Dec 2022, Accepted 19 May 2024, Published online: 24 Jun 2024

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.

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 [Citation1]. Over time, this concept has undergone significant refinement and extension. Notably, Zalmai [Citation2] introduced the transformative ρ(η,θ)-invex functions as a generalization of the invex functions.

The exploration of semistrictly preinvex functions on Rn was initiated by Yang and Li [Citation3], while Rapcsak [Citation4] 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 [Citation5] laid the foundation for convex programming as well as duality conditions, with Pini [Citation6] contributing to the development of invex functions in this context. Addressing vector programming problems on differentiable manifolds, Mititelu [Citation7] 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 [Citation8].

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 [Citation9]. Hanson, Mond, and Rueda established pivotal relationships between KKT points and mathematical programming problems [Citation1, Citation10].

Recent advancements include the introduction of generalized invexity by Caristi et al. [Citation11], the proposal of s-type preinvex functions by Ihsan et al. [Citation12], and the delineation of a new class of preinvex functions by Noor et al. [Citation13]. Additionally, Kiliçman et al. introduced generalized preinvex functions [Citation14], alongside several other notable generalizations proposed by various authors [Citation6, Citation7, Citation15–26].

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 Rn, 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

[Citation27, Citation28]

A non-empty subset X of Rn is known as invex if ∃ a bi-function η:Rn×RnRn such that ∀z1,z2X and ∀δ[0,1], we obtain z2+δη(z1,z2)X.

Definition 2.2

[Citation27, Citation28]

On a non-empty subset X of Rn, a function h:XR is known as preinvex if ∃η:Rn×RnRn, a bi-function, such that ∀z1,z2X and ∀δ[0,1], we obtain h(z2+δη(z1,z2))δh(z1)+(1δ)h(z2).

Definition 2.3

[Citation29]

On a non-empty subset X of Rn, a function h:XR is known as prequasi invex if ∃η:Rn×RnRn, a bi-function, s.t., ∀z1,z2X and ∀δ[0,1], we obtain h(z2+δη(z1,z2))max{h(z1),h(z2)}.

Definition 2.4

[Citation25]

On a non-empty subset X of Rn, a function h:XR is known as strictly prequasi invex if ∃ a bi-function η:Rn×RnRn such that ∀z1,z2X, z1z2 and ∀δ(0,1),we obtain h(z2+δη(z1,z2))<max{h(z1),h(z2)}.

Definition 2.5

[Citation25]

On a non-empty subset X of Rn, a function h:XR is known as semi-strictly prequasi invex if ∃ a bi-function η:Rn×RnRn such that ∀z1,z2X, h(z1)h(z2) and ∀δ(0,1), we obtain h(z2+δη(z1,z2))<max{h(z1),h(z2)}.

The following definitions and remarks are introduced in this paper

Definition 2.6

A non-empty subset X of Rn is said to be w-invex if there exists map η:Rn×RnRn and w:RnRn such that w(z2)+δη(z1,w(z2))X,z1,z2Xandδ[0,1].

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:XR is said to be

  1. w-preinvex if ∀ points z1,z2X, and 0δ1, such that h(w(z2)+δη(z1,w(z2)))δh(z1)+(1δ)h(z2).

  2. w-strictly preinvex if ∀ points z1,z2X, z1z2 and 0<δ<1, such that h(w(z2)+δη(z1,w(z2)))<δh(z1)+(1δ)h(z2).

  3. w-prequasi-invex if ∀ points z1,z2X, and 0δ1, such that h(w(z2)+δη(z1,w(z2)))max{h(z1),h(z2)}.

  4. w-strictly prequasi-invex if ∀ points z1,z2X, z1z2 and 0<δ<1, such that h(w(z2)+δη(z1,w(z2)))<max{h(z1),h(z2)}.

  5. w-semi-strictly prequasi-invex if ∀ points z1,z2X, h(z1)h(z2) and 0<δ<1, such that h(w(z2)+δη(z1,w(z2)))<max{h(z1),h(z2)}.

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 η:[0,)×[0,)R be given by: η(z1,z2)=z1(z22),and let the function w:[0,)R be given by: w(z2)=z2+2.Then, [0,) 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:RR, be defined as: h(z1)=z1+k, where k is a constant.

Let the function w:RR be defined as: w(z2)=z27, and consider η:R×RR as: η(z1,z2)=z1z26 for all z1,z2R. Then, the function h is w-preinvex function w.r.t. η and w.

Example 3.3

Consider h:RR, be defined as: h(z1)=z1+k, where k is a constant.

Consider the function w:RR be defined as: w(z2)=z2, and consider η:R×RR be defined as: η(z1,z2)=z1z2 for all z1,z2R. 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:RR, be defined as: h(z1)=z15. Consider the function w:RR be defined as: w(z2)=z26, and consider η:R×RR be defined as: η(z1,z2)=z1z26 for all z1,z2R. 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,)R, be defined as: h(z1)=z1+k, where k is a constant. let the function w:[0,)R be defined as: w(z2)=0, and consider η:[0,)×[0,)R be defined as: η(z1,z2)=z1+2z2 for all z1,z2[0,). 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,)R, be defined as: h(z1)={11,ifz1[0,11)11,ifz1[11,),let the function w:[0,)R be defined as: w(z2)=z2+11, and consider η:[0,)×[0,)R be defined as: η(z1,z2)=z12+z22+11 for all z1,z2[0,). 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 Rn. h:XR is w-preinvex function w.r.t. η and w if and only if the epigraph of h, epi(h)={(z,α):αRandh(z)α},is w-invex w.r.t. the same η and w.

Proof.

Consider h:XR be a w-preinvex function w.r.t. η and w. Suppose that (z1,α) and (z2,β)epi(h), with z1,z2X, h(z1)α and h(z2)β, for α,βR. Thus, h(w(z2)+δη(z1,w(z2)))δh(z1)+(1δ)h(z2).=δα+(1δ)β,for δ[0,1]which implies that (w(z2)+δη(z1,w(z2)),δα+(1δ)β)epi(h).This shows that epi(h) is w-invex w.r.t. the maps η and w. Conversely, assume that epi(h) is w-invex set w.r.t. the maps η and w. Take z1,z2X. Then (z1,h(z1)),(z2,h(z2))epi(h), it follows that (w(z2)+δη(z1,w(z2)), δh(z1)+(1δ)h(z2))epi(h), δ[0,1],then h(w(z2)+δη(z1,w(z2)))δh(z1)+(1δ)h(z2).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:XR be a w-preinvex function w.r.t. the functions η and w, where X is any non-empty w-invex subset of Rn. Then, the level set Mα={zX,h(z)αandαR}is w-invex w.r.t. the same maps η and w.

Proof.

Let z1,z2Mα. Then z1,z2X, h(z1)α andh(z2)α. Since h is w-preinvex then, we have, ∀δ[0,1], h(w(z2)+δη(z1,w(z2)))δh(z1)+(1δ)h(z2).=δα+(1δ)α,for δ[0,1]=α.This shows that w(z2)+δη(z1,w(z2))X, and the level set Mα 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 Rn. The function h:XR is w-prequasi-invex w.r.t. the same η and w if and only if the level set Mα={zX,h(z)αandαR}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 z1,z2Mα. Then, z1,z2X and max{h(z1),h(z2)}α. Since X is a w-invex set, so, (w(z2)+δη(z1,w(z2))X, δ[0,1].Since h is w-prequasi-invex function, we have h(w(z2)+δη(z1,w(z2)))max{h(z1),h(z2)}α,so (w(z2)+δη(z1,w(z2))Mα.This implies that Mα is a w-invex set w.r.t. the maps η and w.

Conversely, assume that the level set Mα is w-invex w.r.t. the maps η and w. Let z1,z2Mα for max{h(z1),h(z2)}=α. Since X is a w-invex set, we have w(z2)+δη(z1,w(z2))X,implies that h(w(z2)+δη(z1,w(z2)))max{h(z1),h(z2)}=α.This prove that h is a w-prequasi-invex function w.r.t. the functions η and w.

Theorem 3.4

Assume that h:XR is a w-preinvex function w.r.t. the maps η and w, where X is any non-empty w-invex subset of Rn and ν=infzXh(z). Then, the set F={zX:h(z)=ν}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 z1,z2X and δ[0,1]. Since h is a w-preinvex then, we have h(w(z2)+δη(z1,w(z2)))δh(z1)+(1δ)h(z2)=δν+(1δ)ν,for δ[0,1]=ν.This follows that w(z2)+δη(z1,w(z2))F,which implies that F is w-invex w.r.t. the maps η and w.

For the other part, contrarily suppose that h(z1)=h(z2)=ν. Since X is a w-invex set, so w(z2)+δη(z1,w(z2))F, δ[0,1].Also, since h is w-strictly preinvex, we have h(w(z2)+δη(z1,w(z2)))<δh(z1)+(1δ)h(z2)=δν+(1δ)ν,for δ[0,1]=ν.This contradiction the fact that ν=infzXh(z). Therefore, F is singleton.

Theorem 3.5

Let h:XR is a w-preinvex function on X, where X is any non-empty w-invex subset of Rn. Therefore, any local minimum point is also a global minimum point.

Proof.

Let z2 be the local minimum point for h. Then, a neighbourhood N of z2 such that (1) h(z1)h(z2), z1NX.(1) Assuming that z2X is not a global minimum point of h, then ∃ a point zX h(z)<h(z2).Since h is w-preinvex X, ∃η and w such that, ∀δ[0,1], we have (2) h(w(z2)+δη(z,w(z2)))δh(z)+(1δ)h(z2)<δh(z2)+(1δ)h(z2)=h(z2).(2) As w(z2)+δη(z1,w(z2))X for each δ[0,1], ∃ϵ>0 such that w(z2)+δη(z1,w(z2))XN for each δ[0,ϵ), in a contradiction with (Equation1).

Theorem 3.6

Let h:XR is a w-preinvex function on X, where X is any non-empty w-invex subset of Rn. 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:XR is w-preinvex on the w-invex set X, where X is any non-empty w-invex subset of Rn, 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 ∀z1,z2X and δ[0,1] h(w(z2)+δη(z1,w(z2)))δh(z1)+(1δ)h(z2),we have, for any k>0, k(h(w(z2)+δη(z1,w(z2))))k(δh(z1)+(1δ)h(z2)), δ[0,1]from which k(h(w(z2)+δη(z1,w(z2))))δ(kh)(z1)+(1δ)(kh)(z2), δ[0,1].

Theorem 3.8

Let X be a non-empty w-invex subset of Rn and hi:XR, i=1,2,,m, be w-preinvex w.r.t. same η and w. Then, i=1mkihi(z) is w-preinvex w.r.t. same η and w, where ki>0, i=1,2,,m.

Proof.

By hypothesis we have hi(w(z2)+δη(z1,w(z2)))δhi(z1)+(1δ)hi(z2).It follows that kihi(w(z2)+δη(z1,w(z2)))δkihi(z1)+(1δ)kihi(z2).And (i=1mkihi)(w(z2)+δη(z1,w(z2)))δ(i=1mkihi)(z1)+(1δ)(i=1mkihi)(z2).Thus, we achieve the proof.

Theorem 3.9

Let h:XR be a w-preinvex w.r.t. η and w, where X is any non-empty w-invex subset of Rn, let ϕ:RR be an increasing and convex. Then ϕh is w-preinvex w.r.t. η and w.

Proof.

As h is w-preinvex w.r.t. η and w, we obtain ∀z1,z2X,δ[0,1], h(w(z2)+δη(z1,w(z2)))δh(z1)+(1δ)h(z2).Due to the fact that ϕ:RR non-decreasing and convex, we obtain ϕ(h(w(z2)+δη(z1,w(z2))))ϕ(δ(h(z1)+(1δ)h(z2))δϕ(h(z1))+(1δ)ϕ(h(z1)),or ϕh(w(z2)+δη(z1,w(z2))δϕh(z1)+(1δ)ϕh(z1).

Definition 3.1

Let X is any non-empty w-invex subset of Rn. h:XR is said to be w-pre pseudo-invex function if ∃ functions η, w and a function b, positive on domain, such that h(z1)<h(z2)h(w(z2)+δη(z1,w(z2)))h(z2)+δ(δ1)b(z1,z2)for every δ(0,1) and z1,z2X.

Theorem 3.10

Let X be a non-empty w-invex subset of Rn. If h is w-preinvex, then h is w-pre pseudo-invex w.r.t. the same η and w.

Proof.

For h(z1)<h(z2), for every δ(0,1), we can write (h(w(z2)+δη(z1,w(z2))))h(z2)+δ[h(z1)h(z2)]<h(z2)+δ[h(z1)h(z2)]δ2[h(z1)h(z2)],=h(z2)+δ(δ1)[h(z2)h(z1)],where b(z1,z2)=h(z2)h(z1)>0.

Theorem 3.11

Let h:XR is a w-pre pseudo-invex function, where X is any non-empty w-invex subset of Rn. Consequently, any local minimum point is also a global minimum point.

Proof.

Let us assume irrationally that z2 be a local minimum point rather than a global one. Then ∃z1X such that h(z1)<h(z2). According to the w-pre pseudo-invexity definition, which means that, we have ∃b(z1,z2)>0, such that, ∀δ(0,1), h(z2+δη(z1,w(z2)))h(z2)+δ(δ1)b(z1,z2)<h(z2),from which h(z2)>h(w(z2)+δη(z1,w(z2))),  ∀δ(0,1), is contradiction with the assumption.

Theorem 3.12

Assume that η(z1,w(z2))0 whenever z1w(z2). 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 Rn. Consequently, any strict local minimum point is h also a strict global minimum point.

Proof.

Contrarily suppose that z2 be strict local minimum but not global; then ∃zX s.t., h(z2)<h(z). Since h is w-pre quasi-invex, we have h(z2+δη(z,w(z2)))h(z),a contradiction.

Theorem 3.13

Let h:XR be a w-prequasi-invex function w.r.t. η, where X is any non-empty w-invex subset of Rn and w. Suppose that ϕ:RR is a increasing function, then, ϕh 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 ϕh(w(z2)+δη(z1,w(z2)))ϕ(max{h(z1),h(z2)}).ϕh(w(z2)+δη(z1,w(z2)))ϕ(max{h(z1),h(z2)})=max{ϕh(z1),ϕh(z2)},which expresses the fact that the composition function ϕh is w-prequasi-invex.

4. Optimality

Let h:RnR and gi:RnR, i=1,2,,m are w-preinvex functions on Rn. A w-preinvex programming problem is formulated as follows: (3) minh(z)subjecttoX={zRn:gi(z)0,i=1,,m}.(3)

Theorem 4.1

If h:RnR is w-strictly preinvex function on X, where X is any non-empty w-invex subset of Rn, then the global optimal solution of problem (Equation3) is unique.

Proof.

Let z1,z2X be different global optimal solutions of problem (Equation3). Then h(z1)=h(z2). Since h is w-strictly invex then, we have h(w(z2)+δη(z1,w(z2)))<δh(z1)+(1δ)h(z2)=h(z1),h(w(z2)+δη(z1,w(z2)))<h(z1),for each δ(0,1).This contradicts the optimality of z2 for problem (Equation3). Hence, the global optimal solution of problem (Equation3) is unique.

Theorem 4.2

Let h:RnR be w-prequasi-invex function on X, where X is any non-empty w-invex subset of Rn and let γ=minzXh(z). The set Y={zX:h(z)=γ} of optimal solution of problem (Equation3) is w-invex set. If h is w-strictly prequasi-invex, then the set Y is a singleton.

Proof.

Let z1,z2X be different global optimal solutions of problem (Equation3). Then h(z1)=h(z2)=γ.Since h:RnR be w-prequasi-invex function on X, then h(w(z2)+δη(z1,w(z2)))max{h(z1),h(z2)}=γwhich implies that h(w(z2)+δη(z1,w(z2))Y, so Y is w-invex set.

For the other part, on contrary let z1,z2Y, z1z2,  δ(0,1), then h(w(z2)+δη(z1,w(z2))X.

Further, since h is w-strictly prequasi invexity function on X, we have h(w(z2)+δη(z1,w(z2)))<max{h(z1),h(z2)}=γ.This contradicts that γ=minzXh(z) and hence the results follows.

Theorem 4.3

Consider the function h:RnR is w-prequasi-invex on a w-invex subset of Rn. Suppose γ=minzXh(z). Then, the optimal solutions of problem (Equation3) i.e. Y={zX :h(z)=γ} is a w-invex set. If h is w-strictly prequasi-invex on a w-invex subset of Rn. Then w-invex set a singleton set.

Proof.

Let z1,z2Y and δ[0,1], then z1,z2X also h(z1)=γ, h(z2)=γ. As h is w-prequasi-invex function on w-invex set, we have w(z2)+δη(z2,w(z1))Xand h(w(z2)+δη(z2,w(z1)))max{h(z1),h(z2)}=γwhich shows that (w(z2)+δη(z2,w(z1))Y,it demonstrate that w-invex set.

For the other part, suppose to the contrary that z1,z2X, and z1z2, δ(0,1), we get w(z2)+δη(z2,w(z1))X.As h is w-strictly prequasi-invex on X, we get h(w(z2)+δη(z2,w(z1)))<max{h(z1),h(z2)}=γ,there is a contradiction.

Theorem 4.4

Let X be a non-empty w-invex subset of Rn, and let h:RnR be w-strictly prequasi-invex function on X. Also, the inequality h(w(z2)+δη(z2,w(z1)))>h(z2)+2δ[h(z1)h(z2)] holds for all z2X, then the problem (Equation3) has an optimal solution.

Proof.

As h is w-preinvex w.r.t. η and w, we obtain ∀z1,z2X,z1z2, δ(0,1), (4) h(w(z2)+δη(z1,w(z2)))<δh(z1)+(1δ)h(z2).(4) Also, given that (5) h(w(z2)+δη(z2,w(z1)))>h(z2)+2δ[h(z1)h(z2)],for all z2X.(5) Combining the inequalities (Equation4) and (Equation5), we have δ(h(z1)h(z2))<0,for all z2Xso h(z1)h(z2)<0,for all z2X.Hence, the problem (Equation3) has an optimal solution of h.

Theorem 4.5

Let X be a non-empty w-invex subset of Rn w.r.t. η and w, and let h:XR be w-semi-strictly prequasi-invex function for the same η and w. If z¯X is a local optimal solution to the problem of minimizing h(z) subject to zX, then z¯ is a global minimum.

Proof.

Suppose that z¯X is a local minimum. Then, there is an ϵ-neighbourhood Nϵ(z¯) around z¯ such that (6) h(z¯)h(z), zXNϵ(z¯).(6) If z¯ is not a global minimum of h, then there exists an zX such that (7) h(z)<h(z¯).(7) By the w-semi-strictly preinvexity of h, and using (Equation7), we have h(w(z¯)+δη(z,w(z¯)))<δh(z)+(1δ)h(z¯)<δh(z¯)+(1δ)h(z¯)=h(z¯),for all 0<δ<1. For a sufficiently small δ>0, it follows that w(z¯)+δη(z,w(z¯))XNϵ(z¯),which is a contradiction to (Equation6). 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)=L2+R2, 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+R1500,R,L0.

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).

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