Complexity
Volume 2020 (2020), Article ID 9849636, 8 pages
https://doi.org/10.1155/2020/9849636

Research Article

Optimality Conditions for a Nonsmooth Uncertain Multiobjective Programming Problem

Wenyan Han, Guolin Yu, and Tiantian Gong

Institute of Applied Mathematics, North Minzu University, Yinchuan 750021, China

Correspondence should be addressed to Guolin Yu; guolin_yu@126.com

Received 13 April 2020; Accepted 1 July 2020; Published 25 July 2020

Academic Editor: Sigurdur F. Hafstein

Copyright © 2020 Wenyan Han et al. 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.

Abstract.

This note is devoted to the investigation of optimality conditions for robust approximate quasi weak efficient solutions to a nonsmooth uncertain multiobjective programming problem (NUMP). Firstly, under the extended nonsmooth Mangasarian–Fromovitz constrained qualification assumption, the optimality necessary conditions of robust approximate quasi weak efficient solutions are given by using an alternative theorem. Secondly, a class of generalized convex functions is introduced to the problem (NUMP), which is called the pseudoquasi-type-I function, and its existence is illustrated by a concrete example. Finally, under the pseudopseudo-type-I generalized convexity hypothesis, the optimality sufficient conditions for robust approximate quasi weak efficient solutions to the problem (NUMP) are established.

1. Introduction

It is well known that multiobjective programming problems are widely used in the fields of portfolio, resource allocation, and information transfer. In practical problems, most of the objectives or constraints to the optimization model are nonsmooth and are affected by various factors with uncertain information. Therefore, it is a very valuable work to study the nonsmooth uncertain optimization problems. Robust optimization is one of the effective methods to deal with uncertain optimization problems. The robust method is committed to ensuring the worst-case solution which is immunized against the data uncertainty to optimization problems, and for its more details, the reader is referred to [1]. In this paper, the optimality conditions to the nonsmooth uncertain multiobjective programming problem (NUMP) are described by using the robust optimization method.

Convexity and its generalization play an important role in mathematical programming, especially in establishing optimality sufficient conditions of optimization problems. Chuong and Kim [2] presented the generalized convex-affine function based on the Mordukhovich subdifferential for a class of nonsmooth multiobjective fractional programming problems. For objective and constraint functions of a class of nonsmooth robust multiobjective programming problems, the concept that is a generalized convexity of degree is introduced in literature [3]. Inspired by the generalized convexity in the above literatures, this paper introduces a kind of generalized convexities based on the Clarke subdifferential, which is called the -pseudoquasi-type-I function, and under its assumption, it proves the optimality conditions of the problem (NUMP).

As we all know, the (weak) efficient solution of multiobjective optimization problems usually does not exist in the noncompact case, but the approximate solution exists under very mild conditions. In addition, most of the solutions obtained by the numerical algorithm are approximate solutions in the real world. Therefore, there exist the important theoretical value and practical significance to study the approximate solution of the optimization problem. Recently, Lee and Jiao [4] dealt with the optimality conditions of the robust approximate solution for an uncertain convex optimization problem involving a kind of constraint qualifications, which is called closed convex cone constrained qualification; Sun [5] established the optimality conditions of the robust optimal solution under the constrained qualification with respect to the subdifferential of the convex function; Sun and Li [6] discussed the optimality conditions of the robust approximate weak efficient solution under the hypothesis of closed convex cone constrained qualification. It is worth mentioning that the approximate weak efficient solution is a special case of the approximate quasi weak efficient solution. The purpose of this paper is to study optimality conditions of the robust approximate quasi weak efficient solution for the problem (NUMP). The convexities and constrained qualification are different from those of mentioned literatures, and we adopt the newly introduced -pseudopseudo-type-I convexity and extended nonsmooth Mangasarian–Fromovitz constrained qualification (see [7]).

The content of this paper is arranged as follows: In Section 2, some basic concepts and lemmas which will be used in subsequent sections are proposed. The concept of the -pseudoquasi-type-I generalized convexity with respect to the Clarke subdifferential is introduced, and an example is given to illustrate its existence. The main results are present in Section 3, in which the optimality conditions of the robust approximate quasi weak efficient solution to the problem (NUMP) are proven.

2. Preliminaries

This paper considers the following nonsmooth multiobjective programming problem (NMP):

where is a nonempty subset of dimension Euclid space and , are the Lipschitz functions. The feasible set of the problem (NMP) is denoted as

When the constraint set of problem (NMP) contains uncertain data, the corresponding nonsmooth uncertain multiobjective programming problem (NUMP) is expressed as

where is the uncertain parameter, is the compact convex set, and , are the Lipschitz functions with respect to the first variable. We denote . The feasible set of problem (NUMP) is defined by

The optimality conditions of problem (NUMP) will be studied by the robust optimization method in this note. For this purpose, we consider the following robust counterpart (see [8]) of the problem (NUMP):

The robust counterpart problem is called as the nonsmooth robust multiobjective programming problem (NRMP), and the robust feasible set of problem (NRMP) is given by

Let

For a given , we divide into two index sets, , where

For any , let

Let be a closed unit ball in . For any , use to represent the inner product between and . Set

It is said that is a convex function, if for any ,

If is a convex function, then is said to be a concave function. For any , if

then is termed to be a upper semicontinuous function. Let be a nonempty open subset. It is said that is a Lipschitz function, if there exists , such that

Let , and the directional derivative (see [9]) of at in the direction is given by

and the Clarke generalized directional derivative (see [9]) of at in the direction is defined by

If

then is called to be regular at . The Clarke subdifferential (see [9]) of at is denoted as

Lemma 1 (see [7]). Let be a Lipschitz function, then the following conclusions hold:

(i) is a nonempty compact convex set in , and for any , one has ( is Lipschitz constant of ).(ii) is convex, and

(iii)For any , we have

For a given function , suppose satisfies the following assumptions (see [7]):

(i) is upper semicontinuous in .(ii) is a Lipschitz function with respect to the first variable , that is, is a Lipschitz function for any .(iii) is regular with respect to the first variable , that is,

 where is a Clarke generalized directional derivative with respect to the first variable and is the directional derivative with respect to the first variable .(iv)The Clarke subdifferential with respect to the first variable is weak upper semicontinuous in .

Let

If satisfies Assumptions (i)–(iv), then is a Lipschitz function (see [7]). For a given , set

Remark 1. (see [7]). If Assumptions (i)–(iv) are fufilled, is a convex subset and is a concave on ; then,

Lemma 2 (see [9]). Let be a Lipschitz function, . Then,

If are regular at , then is regular at , and .

Lemma 3 (alternative theorem, see [10]). Let be a convex set, and are convex on , if the following system of inequalities

has no solution on , then there exist , not all zero, such that

Definition 1 (see [3]). Let .

(i)It is said that is a robust quasi weak efficient solution of the problem (NUMP), iff is an quasi weak efficient solution of the problem (NRMP), that is,

(ii)It is called that is a robust quasi efficient solution of the problem (NUMP), iff is an quasi efficient solution of the problem (NRMP), that is,

The following generalized convexity is introduced for the objective and constraint functions to the problem (NUMP).

Definition 2. It is said that is a pseudoquasi-type-I function at , if for any , , there exists , such that

If equation (29) takes a strictly inequality, i.e.,

then is called a strictly pseudoquasi-type-I function at .
The following is an example to illustrate the existence of the pseudoquasi-type-I function.

Example 1. Let , be given by

and be defined as

where . Taking , by simple calculation, we can obtain that , . For any , there exists , such that

Hence, is a pseudoquasi-type-I function at .

3. Optimality Conditions

In this section, we begin with establishing the optimality necessary conditions for a robust quasi weak efficient solution to the problem (NUMP) by using the alternative theorem (Lemma 3).

Theorem 1. In the problem (NUMP), if is a robust quasi weak efficient solution of the problem (NUMP), then there exists not all the zero real values , such that

Proof. Firstly, we claim that the following system of inequalities:

has no solution . Otherwise, there exists , such that

Since

we get

Therefore, there exists , for any , and we arrive at

On the other hand, for any ,

Hence, there exists such that , for any . In addition, for any , one has . Noticing that is a Lipschitz function, we know that there exists , for any :

Let , where . Then, for any , it yields that

which contradicts to the fact that is a robust quasi weak efficient solution of the problem (NUMP).
We conclude from Lemma 1 (ii) that

are convex functions. Again by Lemma 3, it can be known that there exists not all zero real values , such that equation (38) holds.
Next, we will examine the optimality necessary conditions of the robust quasi weak efficient solution to the problem (NUMP). For this purpose, we need to introduce the following extended nonsmooth Mangasarian–Fromovitz constrained qualification.

Definition 3. In the problem (NRMP), let . If

then it is called that the problem (NRMP) satisfies extended the nonsmooth Mangasarian–Fromovitz constrained qualification at .

Theorem 2. In the problem (NUMP), suppose that , satisfy Assumptions (i)–(iv), and for any , is a concave function on . If is a robust quasi weak efficient solution to the problem (NUMP), then there exists and , where , , , and , such that

In addition, if the problem (NRMP) satisfies the extended nonsmooth Mangasarian–Fromovitz constrained qualification at , then there exists and , such that equations (52) and (53) hold.

Proof. Since is a robust quasi weak efficient solution to the problem (NUMP), it yields from Theorem 2 that there exists not all zero real values , such that

According to Lemma 1 (iii), we know that

This means that

this is equivalent to

Noticing that , are the nonempty compact convex set in (by Lemma 1 (i)). Hence, it follows from lop-sided minimax theorem [11] that there exists , such that

Again because

we arrive at

Therefore,

In addition, for any , let . Then, from the above equation, we get that

It yields from Remark 1 that there exists , such that (52) and (53) are true.
On the other hand, if the problem (NRMP) satisfies the extended nonsmooth Mangasarian–Fromovitz constrained qualification at and , then there exists , , such that

and the above equation holds according to Lemma 2. Again by Lemma 1 (iii), we have

which contradicts to Definition 3. Hence, this leads to . □
It is said that equations (52) and (53) are robust optimality necessary conditions of the problem (NUMP). We present the following Theorem 3 which is an optimality sufficient condition for the robust -quasi weak efficient solution to the problem (NUMP).

Theorem 3. In the problem (NUMP), supposing that satisfies the robust optimality necessary conditions.

(i)If is a pseudoquasi-type-I function at , then is a robust -quasi weak efficient solution of the problem (NUMP).(ii)If is a strictly pseudoquasi-type-I function at , then is a robust -quasi efficient solution of the problem (NUMP).

Proof. By the given conditions, it follows that the robust optimality necessary conditions hold at , that is,

Therefore, there exists , such that

Let us prove conclusion (i). If is not a robust -quasi weak efficient solution of the problem (NUMP), then there exists , such that

that is,

On the other hand, in equation (66), if , then . Again since , then

Hence,

Because is a pseudoquasi-type-I function at , combining equation (70) with equation (72), we conclude that for , there exists , such that

Noticing that , it holds that

which contradicts to equation (74). Therefore, is a robust -quasi weak efficient solution of the problem (NUMP).
By the similar arguments, we can prove (ii).
Finally, as the end of this article, we give a concrete example to verify Theorem 3.

Example 2. Consider the following nonsmooth robust multiobjective programming (NRMP) problem:

where

and is given by

Taking , it is obvious to get that is a robust -quasi weak efficient solution of the problem (NRMP). It is easy to know that

For any , there exists , such that

It is clear that is a pseudoquasi-type-I function at . Equations (58) and (59) hold for . It yields from Theorem 3 that is a -quasi weak efficient solution of the problem (NRMP).

4. Conclusions

The optimality conditions of the robust approximate quasi weak efficient solution to a nonsmooth uncertain multiobjective programming problem (NUMP) are studied by using the robust optimization method in this note. Firstly, we have introduced -pseudoquasi-type-I functions to the problem (NUMP), and an example is presented to illustrate its existence. Secondly, under the assumptions that the problem (NUMP) satisfies the extended nonsmooth Mangasarian–Fromovitz constrained qualification and pseudoquasi-type-I convexity, optimality conditions of the robust -quasi weak efficient solution are proved.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

This research was supported by the Natural Science Foundation of China, under Grant no. 11861002; Natural Science Foundation of Ningxia, under Grant no. NZ17112; Key Project of North Minzu University, under Grant no. ZDZX201804; Nonlinear analysis and financial optimization research center of North Minzu University.

References

A. Ben-Tal, L. E. Ghaoui, and A. Nemirovski, Robust Optimization, Princeton University Press, Princeton, NY, USA, 2009.

T. D. Chuong and D. S. Kim, “A class of nonsmooth fractional multiobjective optimization problems,” Annals of Operations Research, vol. 244, no. 2, pp. 367–383, 2016.

M. Fakhar, M. R. Mahyarinia, and J. Zafarani, “On approximate solutions for nonsmooth robust multiobjective optimization problems,” Optimization, vol. 150, pp. 1–31, 2019.

J. H. Lee and L. Jiao, “On quasi -solution for robust convex optimization problems,” Optimization Letters, vol. 10, pp. 1–14, 2016.

X. K. Sun, “Characterizations of robust solution for convex optimization problems with data uncertainty,” Acta Mathematica Scientia, vol. 37A, no. 2, pp. 257–264, 2017.

X. K. Sun, X. B. Li, X. J. Long et al., “On robust approximate optimal solutions for uncertain convex optimization and applications to multiobjective optimization,” Pacific Journal of Optimization, vol. 13, no. 4, pp. 621–643, 2017.

G. M. Lee and P. T. Son, “On nonsmooth optimality theorems for robust optimization problems,” Bulletin of the Korean Mathematical Society, vol. 51, no. 1, pp. 287–301, 2014.

M. Fakhar, M. R. Mahyarinia, and J. Zafarani, “On nonsmooth robust multiobjective optimization under generalized convexity with applications to portfolio optimization,” European Journal of Operational Research, vol. 265, no. 1, pp. 39–48, 2018.

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, NY, USA, 1983.

O. L. Mangasarian, Nonlinear Programming, McGraw-Hill, New York, NY, USA, 1969.

J. P. Aubin and I. V. Ekeland, Applied Nonlinear Analysis, John Wiley & Sons, 1984.