Nonsmooth Vector Optimization Problems and Minty Vector Variational Inequalities

The vector optimization problem may have a nonsmooth objective function. Therefore, we introduce the Minty vector variational inequality (Minty VVI) and the Stampacchia vector variational inequality (Stampacchia VVI) defined by means of upper Dini derivative. By using the Minty VVI, we provide a necessary and sufficient condition for a vector minimal point (v.m.p.) of a vector optimization problem for pseudoconvex functions involving Dini derivatives. We establish the relationship between the Minty VVI and the Stampacchia VVI under upper sign continuity. Some relationships among v.m.p., weak v.m.p., solutions of the Stampacchia VVI and solutions of the Minty VVI are discussed. We present also an existence result for the solutions of the weak Minty VVI and the weak Stampacchia VVI.     

Vector variational inequalities and vector optimization problems on Hadamard manifolds

In this paper, we introduce a Minty type vector variational inequality, a Stampacchia type vector variational inequality, and the weak forms of them, which are all defined by means of subdifferentials on Hadamard manifolds. We also study the equivalent relations between the vector variational inequalities and nonsmooth convex vector optimization problems. By using the equivalent relations and an analogous to KKM lemma, we give some existence theorems for weakly efficient solutions of convex vector optimization problems under relaxed compact assumptions.     

Vector variational inequality with pseudoconvexity on Hadamard manifolds

In this paper, we give some properties for nondifferentiable pseudoconvex functions on Hadamard manifolds, and discuss the connections between pseudoconvex functions and pseudomonotone vector fields. Moreover, we study Minty and Stampacchia vector variational inequalities, which are formulated in terms of Clarke subdifferential for nonsmooth functions. Some relations between the vector variational inequalities and nonsmooth vector optimization problems are established under pseudoconvexity or pseudomonotonicity. The results presented in this paper extend some corresponding known results given in the literatures.     

On nonsmooth V-invexity and vector variational-like inequalities in terms of the Michel–Penot subdifferentials

In this paper, we establish some results which exhibit an application for Michel–Penot subdifferential in nonsmooth vector optimization problems and vector variational-like inequalities. We formulate vector variational-like inequalities of Stampacchia and Minty type in terms of the Michel–Penot subdifferentials and use these variational-like inequalities as a tool to solve the vector optimization problem involving nonsmooth V-invex function. We also consider the corresponding weak versions of the vector variational-like inequalities and establish various results for the weak efficient solutions.     

Nonsmooth Multiobjective Problems and Generalized Vector Variational Inequalities Using Quasi-Efficiency

In this paper, a multiobjective problem with a feasible set defined by inequality, equality and set constraints is considered, where the objective and constraint functions are locally Lipschitz. Here, a generalized Stampacchia vector variational inequality is formulated as a tool to characterize quasi- or weak quasi-efficient points. By using two new classes of generalized convexity functions, under suitable constraint qualifications, the equivalence between Kuhn–Tucker vector critical points, solutions to the multiobjective problem and solutions to the generalized Stampacchia vector variational inequality in both weak and strong forms will be proved.     

On vector optimization problems and vector variational inequalities using convexificators

In this paper, we establish some results which exhibit an application of convexificators in vector optimization problems (VOPs) and vector variational inequaities involving locally Lipschitz functions. We formulate vector variational inequalities of Stampacchia and Minty type in terms of convexificators and use these vector variational inequalities as a tool to find out necessary and sufficient conditions for a point to be a vector minimal point of the VOP. We also consider the corresponding weak versions of the vector variational inequalities and establish several results to find out weak vector minimal points.     

On Minty variational principle for nonsmooth vector optimization problems with generalized approximate convexity

In this paper, we consider a nonsmooth vector optimization problem involving locally Lipschitz generalized approximate convex functions and find some relations between approximate convexity and generalized approximate convexity. We establish relationships between vector variational inequalities and nonsmooth vector optimization problem using the generalized approximate convexity as a tool.     

广义向量变分不等式和广义对偶模型

研究实Banach空间中带有不等式约束的非光滑向量优化问题(VP).首先,借助下方向导数引进了广义Minty型向量变分不等式,并通过变分不等式来探讨问题(VP)的最优性条件.接着,利用函数的上次微分构造了不可微向量优化问题(VP)的广义对偶模型,并且在适当的弱凸性条件下建立了弱对偶定理.     

Existence results for Stampacchia and Minty type vector variational inequalities

In this article, we consider the general forms of Stampacchia and Minty type vector variational inequalities for bifunctions and establish the existence of their solutions in the setting of topological vector spaces. We extend these vector variational inequalities for set-valued maps and prove the existence of their solutions in the setting of Banach spaces as well as topological vector spaces. We point out that our vector variational inequalities extend and generalize several vector variational inequalities that appeared in the literature. As applications, we establish some existence results for a solution of the vector optimization problem by using Stampacchia and Minty type vector variational inequalities.     

Generalized Minty Vector Variational-Like Inequalities and Vector Optimization Problems

In this paper, we study the relationship among the generalized Minty vector variational-like inequality problem, generalized Stampacchia vector variational-like inequality problem and vector optimization problem for nondifferentiable and nonconvex functions. We also consider the weak formulations of the generalized Minty vector variational-like inequality problem and generalized Stampacchia vector variational-like inequality problem and give some relationships between the solutions of these problems and a weak efficient solution of the vector optimization problem.     

On Efficiency Conditions for Nonsmooth Vector Equilibrium Problems with Equilibrium Constraints

In this article, necessary conditions of Fritz John type for weak efficient solutions of a nonsmooth vector equilibrium problem involving equilibrium constraints (VEPEC) in terms of the Clarke subdifferentials are established. Under constraint qualifications which are suitable for (VEPEC), necessary conditions of Kuhn-Tucker type for efficiency are derived. Under assumptions on generalized convexity of data, sufficient conditions for efficiency are developed. Some applications to vector variational inequalities and vector optimization problems with equilibrium constraints are also given.     

Existence of Solutions and Variational Principles for Generalized Vector Systems

By means of generalized KKM theory, we prove a result on the existence of solutions and we establish general variational principles, that is, vector optimization formulations of set-valued maps for vector generalized systems. A perturbation function is involved in general variational principles. We extend the theory of gap functions for vector variational inequalities to vector generalized systems and we prove that the solution sets of the related vector optimization problems of set-valued maps contain the solution sets of vector generalized systems. A further vector optimization problem is defined in such a way that its solution set coincides with the solution set of a weak vector generalized system. Research carried on within the agreement between National Sun Yat-Sen University of Kaohsiung, Taiwan and Pisa University, Pisa, Italy, 2007. L.C. Ceng research was partially supported by the National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality grant (075105118). J.C. Yao research was partially supported by the National Science Center for Theoretical Sciences at Tainan.     

Stampacchia向量均衡问题与向量相补问题

引进Stampacchia向量均衡问题与一种新的向量相补问题．用数值方法,得到它们的存在定理,并讨论Stampacchia广义向量变分不等式,向量隐相补问题与极小元问题的关系．     

Gap Functions and Existence of Solutions to Set-Valued Vector Variational Inequalities

The variational inequality problem with set-valued mappings is very useful in economics and nonsmooth optimization. In this paper, we study the existence of solutions and the formulation of solution methods for vector variational inequalities (VVI) with set-valued mappings. We introduce gap functions and establish necessary and sufficient conditions for the existence of a solution of the VVI. It is shown that the optimization problem formulated by using gap functions can be transformed into a semi-infinite programming problem. We investigate also the existence of a solution for the generalized VVI with a set-valued mapping by virtue of the existence of a solution of the VVI with a single-valued function and a continuous selection theorem.     

System of nonsmooth variational inequalities with applications

In this paper, we consider a system of vector variational inequalities and a system of nonsmooth variational inequalities defined by means of Clarke directional derivative. We also consider the Nash equilibrium problem with vector pay-offs and its scalarized form. We present some relations among these systems and problems. The existence results for a solution of system of nonsmooth variational inequalities are given. As a consequence, we derive an existence result for a solution of Nash equilibrium problem with vector pay-offs.     

优化和均衡的等价性

通过向量优化问题, 向量变分不等式问题以及向量变分原理来分析优化问题及均衡问题的一致性.从而显然, 可以用统一的观点来处理数值优化、向量优化以及博弈论等问题.进而为非线性分析提供了一个新的发展空间.     

On Relations Between Vector Optimization Problems and Vector Variational Inequalities

We obtain equivalences between weak Pareto solutions of vector optimization problems and solutions of vector variational inequalities involving generalized directional derivatives.     

On minty variational principle for nonsmooth vector optimization problems with approximate convexity

In this paper, we consider a vector optimization problem involving locally Lipschitz approximately convex functions and give several concepts of approximate efficient solutions. We formulate approximate vector variational inequalities of Stampacchia and Minty type and use these inequalities as a tool to characterize an approximate efficient solution of the vector optimization problem.     

Weak Sharp Solutions for Nonsmooth Variational Inequalities

In this paper, we study the weak sharp solutions for nonsmooth variational inequalities and give a characterization in terms of error bound. Some characterizations of solution set of nonsmooth variational inequalities are presented. Under certain conditions, we prove that the sequence generated by an algorithm for finding a solution of nonsmooth variational inequalities terminates after a finite number of iterates provided that the solutions set of a nonsmooth variational inequality is weakly sharp. We also study the finite termination property of the gradient projection method for solving nonsmooth variational inequalities under weak sharpness of the solution set.     

Decomposition of generalized vector variational inequalities

A multicriteria optimization problem is called Pareto reducible if its weakly efficient solutions actually are efficient solutions for the problem itself or for at least one subproblem obtained from it by selecting certain criteria. The aim of this paper is to investigate a similar property within a special class of generalized vector variational inequalities, under appropriate generalized convexity assumptions.