Optimality and Duality of Approximate Quasi Weakly Efficient Solution for Nonsmooth Vector Optimization Problems

Li, Wenjing; Yu, Guolin

doi:https://doi.org/10.1155/2022/8972971

Mathematical Problems in Engineering

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Recent Developments in Nonlinear Optimization and Control for Large Scale Systems

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Volume 2022 | Article ID 8972971 | https://doi.org/10.1155/2022/8972971

Optimality and Duality of Approximate Quasi Weakly Efficient Solution for Nonsmooth Vector Optimization Problems

Wenjing Li¹and Guolin Yu¹

Academic Editor: S. K. Mishra

Received12 May 2022

Revised19 Jul 2022

Accepted22 Jul 2022

Published15 Sept 2022

Abstract

This paper aims at studying optimality conditions and duality theorems of an approximate quasi weakly efficient solution for a class of nonsmooth vector optimization problems (VOP). First, a necessary optimality condition to the problem (VOP) is established by using the Clarke subdifferential. Second, the concept of approximate pseudo quasi type-I function is introduced, and under its hypothesis, a sufficient optimality condition to the problem (VOP) is also obtained. Finally, the approximate Mond–Weir dual model of the problem (VOP) is presented, and then, weak, strong, and converse duality theorems are established.

1. Introduction

Optimality and duality are two core contents of vector optimization theory and applications. Subdifferential and subgradient (see [1]) as the powerful tools to characterize nonsmooth vector optimization problems have attracted much attention from many scholars (see [2–6]). Clarke subdifferential (see [1, 7, 8]) is an important nonsmooth analytical tool with many good properties. This paper will employ Clarke subdifferential to study the optimality conditions and duality theorems for a class of nonsmooth vector optimization problem (VOP).

In practical problems, most solutions obtained by numerical algorithms are approximate ones. Therefore, it is of great theoretical value and practical significance to study approximate solutions of vector optimization problems. In addition, the convexity of functions and its generalization play a critical role in establishing sufficient conditions for vector optimization problems. Until now, the concepts of invexity (see [9]), pseudo-convexity, and quasi-convexity (see [10]) had been introduced, and the optimality conditions for different efficient solutions to vector optimization problems had been discussed. For more references dealing with approximate solutions and approximate convexity, please refer to [11–14]. In this paper, we will propose a kind of generalized convexity, termed as the approximate pseudo quasi type-I functions, and use it to explore optimality to the problem (VOP) with respect to the approximate quasi weakly efficiency.

It is known that duality assertions allow to study a minimization problem through a maximization problem and to know what one can expect in the best case. There are many dual models in the literature, for instance, Lagrange dual (see [15, 16]), Mond–Weir duality (see [17–19]), Wolfe duality (see [17, 20]), conjugate duality (see [21]), and symmetric duality (see [22]). In this paper, we will introduce an approximate Mond–Weir dual model for the problem (VOP) and examine duality theorems between it and primal problem involving approximate quasi weakly efficient solution.

The article is organized as follows: in Section 2, we give some symbols, concepts, and lemmas, which will be used in the subsequent sections; in Section 3, the necessary and sufficient optimality conditions are established; in Section 4, an approximate Mond–Weir dual model of the problem (VOP) is presented, and then, some duality theorems are obtained.

2. Preliminaries

Let be the -dimensional Euclidean space, is a nonempty subset of , , , and denote its interior, closure, and convex hull, respectively. stands for the open ball of radius around . The inner product between and is denoted by for any , and the norm of be defined by . Let

Without other specifications, we always suppose that be pointed closed convex cone with . The dual cone of be denoted by (see [1])

Definition 1 (see [3, 23]). An vector-valued mapping is said to be locally Lipschitz at , if there exists a constant and such thatIf is locally Lipschitz at , for any , then is called locally Lipschitz mapping on .
It had been pointed out in [24] that the sum of two locally Lipschitz mappings is locally Lipschitz in finite-dimensional spaces.

Definition 2 (see [1]). Let be a locally Lipschitz function, the Clarke directional derivative of at in the direction be defined asThe subdifferential of at is defined as

Definition 3 (see [1]). Let be a nonempty subset of , the Clarke contingent cone and Clarke normal cone associated with a set at a point are defined by

Lemma 1 (see [24]). Let be a locally Lipschitz at and the function be a locally Lipschitz at , then

Definition 4 (see [25]). Let be a nonempty subset of , the indicator function on is defined as

Remark 1. If is a closed set, then is a lower semicontinuous function.
We collect some properties of Clarke subdifferential in the following Lemma 2, which will be used in the later subsection, and their proofs can be found in [1] pp. 25–109.

Lemma 2 (see [1]). Let , , and be a locally Lipschitz function. Then,(i)for all , (ii)(iii)if is local minimum point of , then (iv)if is closed set, then

Gerstewitz’s function is an important nonlinear scalarization function (see [26]), which has been proved to be an effective tool for characterizing vector optimization problem. We shall use some properties of Gerstewitz’s function (see pp.302–310 in [26]) to prove optimality conditions in Section 3.

Lemma 3 (see [26]). Let , Gerstewitz’s function be defined asand is continuous and locally Lipschitz.

The next Lemma 4 collects some properties of Gerstewitz’s function (see Lemma 2.3 and Lemma 3.1 in [25]), which will be used to prove the main conclusions of this article.

Lemma 4 (see [25]). Let and . Then,(i)(ii)(iii)(iv)

Let and be vector-valued mappings, be a nonempty closed subset of . We consider the following vector optimization problem (VOP):

Let be the set of feasible solutions for problem (VOP).

The notion of approximate quasi weakly efficiency was introduced in literature [27] (see Definition 3.2). Now, let’s present this concept.

Definition 5 (see [27]). Let , . is called an -quasi weakly efficient solution to problem (VOP), ifNow, we give an example of -quasi weakly efficient solution for problem (VOP).

Example 1. Let , , the vector-valued mapping be defined asand the mapping be defined byTaking , , and , we get , and for all ,Hence, is an -quasi weakly efficient solution of problem (VOP).

3. Optimality Conditions

In this section, we first give a necessary optimality condition using Clarke subdifferential for problem (VOP) with respect to quasi weakly efficiency. Furthermore, a kind of generalized convexity is introduced and is applied to establish a sufficient condition to problem (VOP).

Theorem 1. In problem (VOP), let , , , and be locally Lipschitz on . If is an -quasi weakly efficient solution of problem (VOP), then there exists such thatwhere , and is the closed unit ball of .

Proof. Since is an -quasi weakly efficient solution of problem (VOP), we getThis means thatBy Lemma 4 (ii), we obtainLet , thenParticularly, it holdsNoticing that and , we get from Lemma 4 (i) thatBy (21), we arrive atTogether with (20), we deriveIt means that is a minimal solution of on . Furthermore, this leads to that is a minimal solution of the function on . It yields from Lemma 2 thatSince is closed, it follows from Lemma 2 (iv) thatCombining with (25), we getBecause , , Gerstewitz’s funtion , and are locally Lipschitz, it follows from Lemma 1 that is also locally Lipschitz andThus, there exists with such thatBy Lemma 4 (iii), we obtainHence, . Noticing , we derive from (29) thatwhich implies thatWe now prove that (16) holds. It is clear thatBy Lemma 4 (iv), we obtainthat is,Therefore, (16) is true.
Motivated by the concepts of approximate pseudoconvex of type I and type II given in [10] (see Definition 8 and Definition 9 in [10]), we present the following generalized convexity, which will be utilized to establish a sufficient condition for problem (VOP).

Definition 6. Let , be locally Lipschitz functions. It is said that is -pseudo quasi type-I function at , if for any , , it holds

Example 2. Let , the real-valued functions , be defined byLet , , because functions and are differentiable, we derive , , and for any , and , it holdsHence, is 3-pseudo quasi type-I at .

Theorem 2. In problem (VOP), let , , and . Suppose that and are locally Lipschitz on , and there exist and such that (15) and (16) hold. If is -pseudo quasi type-I at , then is an -quasi weakly efficient solution of problem (VOP).

Proof. It follows from (15) that there exist , , , such thatThis means thatIt is obvious thatTogether with (40), we obtainNoticing that , we arrive atBy (42), one getsSuppose that is not -quasi weakly efficient solution of problem (VOP), there exists such thatBecause , one hasthat is,Since and , we get from (16) thatIn addition, because is -pseudo quasi type-I at , it yieldsHence,which contradicts (43). Thus, is an -quasi weakly efficient solution of problem (VOP).

4. Approximate Mond-Weir Dual

In this section, we propose an approximate Mond–Weir dual model for problem (VOP) and establish duality theorems with respect to quasi weakly efficient solution. Let and , for problem (VOP), we consider the following approximate Mond–Weir dual problem :

The feasible set of problem is denoted by

Definition 7. is called an -quasi weakly efficient solution to problem , if for any , it holdsFeng and Qiu ([25]) proposed a kind of generalized invex functions (see Definition 2.2 in [25]) and used it to derive a sufficient condition for an vector equilibrium problem with respect to weakly efficient solution. We extend the generalized invexity introduced in [25] to the following two class concepts of generalized convex functions and employ them to establish duality theorems between problem (VOP) and .

Definition 8. Let and , be a locally Lipschitz function.(1) is said to be quasi--convex at , if for any , it holds that(2) is said to be -strictly -quasi convex at , if for any , it holds thatThe next Example 3 is an approximate strictly -quasi convex function.

Example 3. Let , , and , be defined asSetting , , because is differentiable, we derive , and for any , , it holdsSinceand for any ,we get that is -strictly -quasi convex at .

Theorem 3 (Weak duality). Let and , . If is -strictly quasi--convex at , is quasi--convex at , then

Proof. Since , we haveIt yields that there exist , , , and such thatThis means thatIt is clear thatThis leads toNoticing that , we obtainwhich implies thatSuppose to the contrary, that is, (60) is not true, and there exists such thatBecause , one getswhich means thatIn view of and , we getSince is -strictly quasi--convex at , and , it yields thatNoticing that , , , and , one hasFrom and , we getBecause and is quasi--convex at , it follows thatCombining with (72) and (75), we arrive atwhich contradicts (67).

Theorem 4 (Strong duality). Let , . Suppose that is an -quasi weakly efficient solution of (VOP), and there exists such that (15) and (16) hold. If is -strictly--convex at , is quasi--convex at , then is an -quasi weakly efficient solution of dual problem .

Proof. Since is an -quasi weakly efficient solution of (VOP), it follows from Theorem 1 that there exists such thatWe claim that , . In fact, if , thendue to . This is a contradiction. Thus, . In addition, since is -strictly quasi--convex at and is quasi--convex at , we get from Theorem 3 thatTherefore, is an -quasi weakly efficient solution of dual problem .

Theorem 5 (Converse duality). Let , , , and there exists such that . If is -strictly quasi--convex at , and are quasi--convex at , then is an -quasi weakly efficient solution of problem (VOP).

Proof. Supposing that is not an -quasi weakly efficient solution of problem (VOP), then there exists such thatSince , it follows thatBecause and , we getBy the similar arguments as in Theorem 3, we concludewhich contradicts to the following equation:Thus, is an -quasi weakly efficient solution of problem (VOP).

5. Conclusions

We have derived a necessary optimality condition for a class of nonsmooth vector optimization problem (VOP) by using the properties of Gerstewitz’s function and Clarke subdifferential. The concept of approximate pseudo quasi type-I function is introduced, and it is utilized to establish a sufficient optimality condition for an approximate quasi weakly efficient solution to problem (VOP). We have also proposed an approximate Mond–Weir dual model for problem (VOP) and established the weak, strong, and converse duality theorems with respect to approximate quasi weakly efficiency.

The optimality of vector optimization problems with respect to approximate solution is an interesting topic. There were some results related to approximate convexities and quasi efficiency in literature [12, 28, 29], and the readers can refer to them and references [30] therein.

Data Availability

The data used to support the findings of this study are available from the author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

Acknowledgments

This research was supported by the Natural Science Foundation of China (No. 11861002); the Key Project of North Minzu University (No. ZDZX201804); and the Undergraduates Project of North Minzu University (No. YCX22109).

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Copyright © 2022 Wenjing Li and Guolin Yu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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