Weak efficiency in multiobjective variational problems under generalized convexity

In this paper, we provide new pseudoinvexity conditions on the involved functionals of a multiobjective variational problem, such that all vector Kuhn-Tucker or Fritz John points are weakly efficient solutions if and only if these conditions are fulfilled. We relate weakly efficient solutions to optimal solutions of weighting problems. We improve recent papers, and we generalize pseudoinvexity conditions used in multiobjective mathematical programming, so as some of their characterization results. The new conditions and results are illustrated with an example.     

Strictly feasible solutions and strict complementarity in multiple objective linear optimization

Recently, Luc defined a dual program for a multiple objective linear program. The dual problem is also a multiple objective linear problem and the weak duality and strong duality theorems for these primal and dual problems have been established. Here, we use these results to prove some relationships between multiple objective linear primal and dual problems. We extend the available results on single objective linear primal and dual problems to multiple objective linear primal and dual problems. Complementary slackness conditions for efficient solutions, and conditions for the existence of weakly efficient solution sets and existence of strictly primal and dual feasible points are established. We show that primal-dual (weakly) efficient solutions satisfying strictly complementary conditions exist. Furthermore, we consider Isermann’s and Kolumban’s dual problems and establish conditions for the existence of strictly primal and dual feasible points. We show the existence of primal-dual feasible points satisfying strictly complementary conditions for Isermann’s dual problem. Also, we give an alternative proof to establish necessary conditions for weakly efficient solutions of multiple objective programs, assuming the Kuhn–Tucker (KT) constraint qualification. We also provide a new condition to ensure the KT constraint qualification.     

Optimality conditions for various efficient solutions involving coderivatives: From set-valued optimization problems to set-valued equilibrium problems

We present a new approach to the study of a set-valued equilibrium problem (for short, SEP) through the study of a set-valued optimization problem with a geometric constraint (for short, SOP) based on an equivalence between solutions of these problems. As illustrations, we adapt to SEP enhanced notions of relative Pareto efficient solutions introduced in set optimization by Bao and Mordukhovich and derive from known or new optimality conditions for various efficient solutions of SOP similar results for solutions of SEP as well as for solutions of a vector equilibrium problem and a vector variational inequality.We also introduce the concept of quasi weakly efficient solutions for the above problems and divide all efficient solutions under consideration into the Pareto-type group containing Pareto efficient, primary relative efficient, intrinsic relative efficient, quasi relative efficient solutions and the weak Pareto-type group containing quasi weakly efficient, weakly efficient, strongly efficient, positive properly efficient, Henig global properly efficient, Henig properly efficient, super efficient and Benson properly efficient solutions. The necessary conditions for Pareto-type efficient solutions and necessary/sufficient conditions for weak Pareto-type efficient solutions formulated here are expressed in terms of the Ioffe approximate coderivative and normal cone in the Banach space setting and in terms of the Mordukhovich coderivative and normal cone in the Asplund space setting.     

Scalarization and optimality conditions for vector equilibrium problems

In this paper, we investigated vector equilibrium problems and gave the scalarization results for weakly efficient solutions, Henig efficient solutions, and globally efficient solutions to the vector equilibrium problems without the convexity assumption. Using nonsmooth analysis and the scalarization results, we provided the necessary conditions for weakly efficient solutions, Henig efficient solutions, globally efficient solutions, and superefficient solutions to vector equilibrium problems. By the assumption of convexity, we gave sufficient conditions for those solutions. As applications, we gave the necessary and sufficient conditions for corresponding solutions to vector variational inequalities and vector optimization problems.     

Optimality conditions for nonsmooth semi-infinite multiobjective programming

This paper is devoted to the study of nonsmooth multiobjective semi-infinite programming problems in which the index set of the inequality constraints is an arbitrary set not necessarily finite. We introduce several kinds of constraint qualifications for these problems, and then necessary optimality conditions for weakly efficient solutions are investigated. Finally by imposing assumptions of generalized convexity we give sufficient conditions for efficient solutions.     

Optimality conditions for vector equilibrium problems with constraint in Banach spaces

In this paper, vector equilibrium problems with constraint in Banach spaces are investigated. Kuhn–Tucker-like conditions for weakly efficient solutions are given by using the Gerstewitz’s function and nonsmooth analysis. Moreover, the sufficient conditions of weakly efficient solutions are established under the assumption of generalized invexity. As applications, necessary conditions of weakly efficient solutions for vector variational inequalities with constraint and vector optimization problems with constraint are obtained.     

多目标优化问题鲁棒有效解与真有效解之间的关系

在一定条件下研究了多目标优化问题鲁棒有效解与真有效解之间的关系及鲁棒有效解的最优性条件.首先,给出多目标优化问题鲁棒弱有效解的概念,研究它与鲁棒有效解和真有效解之间的关系,举例说明了相关结果的合理性.其次,在次类凸和伪凸性假设下研究了鲁棒有效解的必要性条件和充分性条件.     

First and Second Order Optimality Conditions Using Approximations for Nonsmooth Vector Optimization in Banach Spaces

We use the first and second order approximations of mappings to establish both necessary and sufficient optimality conditions for unconstrained and constrained nonsmooth vector optimization problems. Ideal solutions, efficient solutions, and weakly efficient solutions are considered. The data of the problems need not even be continuous. Some often imposed compactness assumptions are also relaxed. Examples are provided to compare our results and some known recent results.This work was partially supported by the National Basic Research Program in Natural Sciences of Vietnam.     

非强制混合向量变分不等式解的存在性研究

本文利用例外簇方法研究非强制混合向量变分不等式的弱有效解的存在性:首先证明若混合向量变分不等式问题不存在例外簇,则混合向量变分不等式问题的弱有效解集为非空集合:利用向量值映射的渐近映射给出自反Banach空间中非强制混合向量变分不等式的弱有效解集不存在例外簇的充分条件,从而得到混合向量变分不等式问题的弱有效解的存在性结...     

Nonsmooth Semi-infinite Multiobjective Optimization Problems

We apply some advanced tools of variational analysis and generalized differentiation to establish necessary conditions for (weakly) efficient solutions of a nonsmooth semi-infinite multiobjective optimization problem (SIMOP for brevity). Sufficient conditions for (weakly) efficient solutions of a SIMOP are also provided by means of introducing the concepts of (strictly) generalized convex functions defined in terms of the limiting subdifferential of locally Lipschitz functions. In addition, we propose types of Wolfe and Mond–Weir dual problems for SIMOPs, and explore weak and strong duality relations under assumptions of (strictly) generalized convexity. Examples are also designed to analyze and illustrate the obtained results.     

Asplund空间中非凸向量均衡问题近似解的最优性条件

在Asplund空间中,研究了非凸向量均衡问题近似解的最优性条件.借助Mordukhovich次可微概念,在没有任何凸性条件下获得了向量均衡问题εe-拟弱有效解,εe-拟Henig有效解,εe-拟全局有效解以及εe-拟有效解的必要最优性条件.作为它的应用,还给出了非凸向量优化问题近似解的最优性条件.     

Existence and stability of solutions for a class of generalized vector equilibrium problems

In this paper, two existence theorems concerning the strong efficient solutions and the weakly efficient solutions of generalized vector equilibrium problems are derived by using the Fan-KKM Theorem and an existence theorem for the efficient solutions of generalized vector equilibrium problems is established by using the scalarization method. Moreover, the lower semicontinuity of the strong efficient solution mapping and the weakly efficient solution mapping to parametric generalized vector equilibrium problems are showed under suitable conditions with neither monotonicity nor any information of the solution mappings. Finally, some applications to the vector optimization problems and the Stackelberg equilibrium problem are also given.     

Merit functions in vector optimization

We study the weak domination property and weakly efficient solutions in vector optimization problems. In particular scalarization of these problems is obtained by virtue of some suitable merit functions. Some natural conditions to ensure the existence of error bounds for merit functions are also given. This research was supported by a direct grant (CUHK) and an Earmarked Grant from the Research Grant Council of Hong Kong.     

Lagrangian conditions for vector optimization in Banach spaces

We consider vector optimization problems on Banach spaces without convexity assumptions. Under the assumption that the objective function is locally Lipschitz we derive Lagrangian necessary conditions on the basis of Mordukhovich subdifferential and the approximate subdifferential by Ioffe using a non-convex scalarization scheme. Finally, we apply the results for deriving necessary conditions for weakly efficient solutions of non-convex location problems.     

Weakly and properly nonessential objectives in multiobjective optimization problems

In this work we characterize objective functions which do not change the set of efficient solutions (weakly efficient solutions, properly efficient solutions). Necessary and sufficient conditions for an objective function to be weakly nonessential (properly nonessential) are presented. We establish relations between weakly nonessential, properly nonessential and nonessential functions.     

Optimality conditions for directionally differentiable multi-objective programming problems

This paper is concerned with the optimality for multi-objective programming problems with nonsmooth and nonconvex (but directionally differentiable) objective and constraint functions. The main results are Kuhn-Tucker type necessary conditions for properly efficient solutions and weakly efficient solutions. Our proper efficiency is a natural extension of the Kuhn-Tucker one to the nonsmooth case. Some sufficient conditions for an efficient solution to be proper are also given. As an application, we derive optimality conditions for multi-objective programming problems including extremal-value functions.This work was done while the author was visiting George Washington University, Washington, DC.     

Duality and Weak Efficiency in Vector Variational Problems

We establish weak, strong, and converse duality results for weakly efficient solutions in vector or multiobjective variational problems, which extend and improve recent papers. For this purpose, we consider Kuhn–Tucker optimality conditions, weighting variational problems, and some classes of generalized convex functions, recently introduced, which are extended in this work. Furthermore, a related open question is discussed.     

Second-Order Optimality Conditions for Set-Valued Vector Equilibrium Problems

In this article, by using the generalized second-order contingent (adjacent) epiderivatives of set-valued maps, we obtain necessary optimality conditions and sufficient optimality conditions for weakly efficient solutions, Henig efficient solutions to the set-valued vector equilibrium problems with constraints. Some results of this article improve the corresponding results in literatures by lessening the assumption of convexity.     

Vector variational-like inequalities and non-smooth vector optimization problems

In this paper, we establish relationships between vector variational-like inequality problems and non-smooth vector optimization problems under non-smooth invexity. We identify the vector critical points, the weakly efficient points and the solutions of the non-smooth weak vector variational-like inequality problems, under non-smooth pseudo-invexity assumptions. These conditions are more general than those existing in the literature.     

Necessary and sufficient second order optimality conditions for multiobjective problems with C1 data

In this paper, we obtain necessary and sufficient second order optimality conditions for multiobjective problems using second order directional derivatives. We propose the notion of second order KT-pseudoinvex problems and we prove that this class of problems has the following property: a problem is second order KT-pseudoinvex if and only if all its points that satisfy the second order necessary optimality condition are weakly efficient. Also we obtain second order sufficient conditions for efficiency.