ZHANG Ya-meng, YU Guo-lin

(Institute of Applied Mathematics, North Minzu University, Yinchuan 750021)

Abstract: This paper is devoted to the study of the relations between vector variational inequality and nonsmooth multi-objective optimization in the sense of strict minimizers of higher order. We firstly introduce an extension of higher-order strong pseudoconvexity for Lipschitz functions, termed higher-order strongly pseudoconvex functions of type I, and some examples are presented in the support of this generalization. Then, we identify the strict minimizers of higher order, the vector critical points and the solutions of the weak vector variational inequality problem under the higher-order strong pseudoconvexity of type I hypothesis. It is our understanding that such results have not been established till now.

Keywords: multi-objective optimization; strict minimizer of higher order; vector variational inequality; strong convexity

1 Introduction

In optimization, the notion of strict minimizer of higher order plays an important role in the convergence analysis of numerical methods and in stability results,the interest toward this kind of solutions has grown. In this work,we focus on strict minimizers of higher order for nonsmooth multi-objective optimization problems. For more details about strict minimizer and its applications in the scalar optimization, we refer Auslender[1], Ward[2]and Sahay and Bhatia[3]. Bhatia[4]and Jim′enez[5]defined the concept of strict minimizer of higher order for vector optimization and multi-objective optimization,respectively. Bhatia and Sahay[6]introduced the notion of strict minimizers of higher order with respect to a nonlinear function and examined the optimality conditions and duality theorems for a differentiable multi-objective optimization problem. Recently,Yu[7]dealt with the optimality conditions for strict minimizers of higher order for a nonsmooth semi-infinite multi-objective optimization problem.

On the other hand, it is well known that convexity and its generalization are of great importance in the field of optimization. As a meaningful generalized convexity,the strongly convex function, which was firstly introduced by Lin and Fukushima[8],has been investigated by several scholars[4,6-8]. Here, it is worth emphasizing the work of Bhatia in [4]. Bhatia introduced a weak vector variational inequality problem, and presented that its solution is equivalent to the strict minimizer of higher order for the involved multi-objective optimization under the assumption of higher-order strong convexity. One contribution of this note is to extend this result to the case of more general higher-order strong convexity. In addition, we also pay attention to the relations between the strict minimizer of higher order and vector critical point in a multi-objective optimization problem. In fact, some recent works have shown that for some generalized convex objective functions, a point is a (weak) efficient solution if and only if it is a vector critical point. For example, Arana-Jim′enezet al[9]used the strong pseudoinvexity for a differentiable function; Santoset al[10]employed pseudoinvexity for a Fr′echet differentiable function;Mishra and Upadhyay[11]were interested in approximate pseudoconvexity for a Lipschitz function; Guti′errezet al[12]focused on strong pseudoconvexity defined through the generalized Jacobian, and so on. The other contribution of this work is to prove that for a higher-order generalized strong pseudoconvex Lipschitz objective function, a point is a strict minimizer of higher order if and only if it is a vector critical point. Based upon above mentioned two contributions, we can identify the vector critical points, the higher-order strict minimizer and the solutions of the weak vector variational inequality problem under higher-order extended strong pseudoconvexity assumptions.

The paper is organized as follows. At the beginning of section 2, we specify the main notations and we recall some basic definitions needed in the sequel. After that,we introduce a new extension of higher-order strong pseudoconvexity for a Lischitz vector valued function, named higher-order strongly pesudoconvex function of type I. Examples are provided in the support of this generalization. In section 3, we present certain relations between a nonsmooth multi-objective optimization problem and a weak vector variational inequality problem by using the concepts of higher-order strong pseudoconvexity of type I and strict minimizer of higher order hypothesis. We also distinguish the vector critical points, the strict minimizer of higher order to the nonsmooth multiobjective optimization and the solutions of the weak vector variational inequality.

2 Notations and preliminaries

Let Rnbe then-dimensional Euclidean space endowed with the Euclidean norm‖·‖andbe the nonnegative orthant of Rn. As usual, we use int(A) to denote the interior of a setA. Throughout of the paper, we always assume thatXis a nonempty open convex subset of Rnandm ≥1 is a positive integer, and adopt the following conventions for vectors in Rn.

Recall that a functionφ:X →R is Lipschitz nearx ∈X, if there exists a positive constantLsuch that

φis called to bem-order strongly convex onXif it ism-order strongly convex at everyx ∈X.

or equivalently

φis called to bem-order strongly pseudoconvex function of type I onXifφism-order strongly pseudoconvex function of type I at eachx ∈X.

It is evident thatm-order strong convexity impliesm-order strong pseudoconvexity of type I. However, the converse is not true in general. We illustrate this fact by the following example.

Example 1 Consider the following function:φ:R→R, defined by

does not hold for anyc ＞0.

Based upon Definition 4,we define the vector valuedm-order strong pseudoconvex function of type I as follows.

We present an example to illustrate the existence ofm-order strongly pseudoconvex vector valued function of type I.

Example 2 Consider the functionf:R→R2,f(x)=(f1(x),f2(x)) forx ∈R,defined as

wherefi,i=1,2,··· ,pare Lipschitz fromXto R.

Let us present the notion of strict minimizer of ordermfor (NMOP), which was firstly defined by Jim′enez[5].

Associated with the problem (NMOP), Bhatia[4]presented the following weak vector variational inequality problem:

3 Relationships with multi-objective optimization

In this section, by employing the tools of nonsmooth analysis and the concept of strict minimizer of higher order,we examine the relations between the nonsmooth multiobjective optimization problem (NMOP) and the weak vector variational inequality problem (WVVIP).

This is a contradiction to the fact that ˉxis a strict minimizer of ordermto the(NMOP).

The following result can be directly obtained from Theorem 1 and Theorem 2.

Corollary 1 Supposefi, i=1,2,··· ,p, are regular andm-order strongly pseudoconvex functions of type I onX. Then ˉxsolves the (WVVIP) if and only ifis a strict minimizer of ordermto the (NMOP).

Remark 4 It is observed from Remark 2 and Remark 3 that Corollary 1 extends Theorem 5.1 in [4].

The concept of vector critical point for a multi-objective optimization problem involving differentiable functions is presented by Osuna-G′omezet al[15]. Now,we extend this concept to the nonsmooth case.

The following lemma is the famous Gordan theorem(see[16]),which plays a critical role in proving our main results.

Lemma 1[16]LetA ∈Rp×nbe a given matrix. Then,exactly one of the following systems is consistent:

(I) There existsx ∈Rnsuch thatAx ＜0;

(II) There existsy ∈Rpwithy≥0 such thatATy=0, whereATis the transposition ofA.

has no solution forβ, this leads to a contradiction to the equation (4).

Theorem 4 Any vector critical point is a strict minimizer of ordermto the(NMOP),if and only iff:X →Rpis am-order strongly pseudoconvex type I function at that point.

Proof The sufficiency yields from Theorem 3. It is only needed to prove that if every vector critical point is a strict minimizer of ordermto the (NMOP), then the vector valued functionfsatisfies the condition form-order strongly pseudoconvexity of type I at that point.

This shows thatfism-order strongly pseudoconvex function of type I at ˉx.

In the light of Corollary 1 and Theorem 4, we can establish a characterization of the vector critical points through the solutions to the (WVVIP) with the following result.

Corollary 2 Iffi, i=1,2,··· ,p,are regular andm-order strongly pseudoconvex functions of type I onX, then the vector critical points, the strict minimizer of ordermto the (NMOP), and the solutions of (WVVIP) are equivalent.

4 Conclusions

We have introduced a new generalization of higher-order strong pseudoconvexity for the Lischitz functions, calledm-order strongly pesudoconvex functions of type I.Examples are presented to illustrate its existence. We have examined the relations between vector variational inequality problems and nonsmooth multi-objective optimization problems in the sense of strict minimizer of ordermunder them-order strong pseudoconvexity of type I assumption. We have also obtained that the vector critical points, the strict minimizers of ordermto a nonsmooth multi-objective optimization problem and the solutions of a weak vector variational inequality problem are equivalent. The results of this note extend some earlier results of Bhatia to a more general class of functions.