Second-Order Optimality Conditions in Multiobjective Optimization Problems

In this paper, we develop second-order necessary and sufficient optimality conditions for multiobjective optimization problems with both equality and inequality constraints. First, we generalize the Lin fundamental theorem (Ref. 1) to second-order tangent sets; then, based on the above generalized theorem, we derive second-order necessary and sufficient conditions for efficiency.     

On Constraint Qualification in Multiobjective Optimization Problems: Semidifferentiable Case

Some versions of constraint qualifications in the semidifferentiable case are considered for a multiobjective optimization problem with inequality constraints. A Maeda-type constraint qualification is given and Kuhn–Tucker-type necessary conditions for efficiency are obtained. In addition, some conditions that ensure the Maeda-type constraint qualification are stated.     

New Results on Second-Order Optimality Conditions in Vector Optimization Problems

In this paper, we study second-order optimality conditions for multiobjective optimization problems. By means of different second-order tangent sets, various new second-order necessary optimality conditions are obtained in both scalar and vector optimization. As special cases, we obtain several results found in the literature (see reference list). We present also second-order sufficient optimality conditions so that there is only a very small gap with the necessary optimality conditions. The authors thank Professor P.L. Yu and the referees for valuable comments and helpful suggestions.     

非光滑非凸多目标规划解的充分条件

Kuhn-Tucker型条件的充分性一直是最优化理论中引人注意的一个问题.本文对非光滑函数提出了几个非凸概念,然后,讨论了非光滑非凸多目标规划中Kuhn-Tucker型条件和Fritz John型条件的充分性,在很弱的条件下,建立了一系列充分条件.     

Optimality Conditions for Isolated Local Minimum of Nonsmooth Multiobjective Problems

This article is devoted to the study of Fritz John and strong Kuhn-Tucker necessary conditions for properly efficient solutions, efficient solutions and isolated efficient solutions of a nonsmooth multiobjective optimization problem involving inequality and equality constraints and a set constraints in terms of the lower Hadamard directional derivative. Sufficient conditions for the existence of such solutions are also provided where the involved functions have pseudoconvex sublevel sets. Our results are based on the concept of pseudoconvex sublevel sets. The functions with pseudoconvex sublevel sets are a class of generalized convex functions that include quasiconvex functions.     

Optimality Criteria for Nonsmooth Continuous-Time Problems of Multiobjective Optimization

A nonsmooth multiobjective continuous-time problem is introduced. We establish the necessary and sufficient optimality conditions under generalized convexity assumptions on the functions involved. This research was supported by Center of Excellence for Mathematics, University of Isfahan, Isfahan, Iran.     

Optimality Conditions for Nonsmooth Multiobjective Optimization Using Hadamard Directional Derivatives

We establish both necessary and sufficient optimality conditions for weak efficiency and firm efficiency by using Hadamard directional derivatives and scalarizing the multiobjective problem under consideration via signed distances. For the first-order conditions, the data of the problem need not even be continuous; for the second-order conditions, we assume only that the first-order derivatives of the data are calm. We include examples showing the advantages of our results over some recent papers in the literature. This work was partially supported by the National Basic Research Program in Natural Sciences of Vietnam. The authors are indebted to Professor P.L. Yu and two anonymous referees for many valuable remarks, which helped improving the previous version of the paper.     

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.     

Optimality Conditions for C1,1 Vector Optimization Problems

The main purpose of this paper is to make use of the second-order subdifferential of vector functions to establish necessary and sufficient optimality conditions for vector optimization problems.     

Sufficiency in Nonsmooth Multiobjective Programming Involving Generalized (Fρ)-convexity

In this paper, we consider a generalization of convexity for nonsmooth multiobjective programming problems. We obtain sufficient optimality conditions under generalized (Fρ)-convexity.This work was supported by Project 821134 and by the Center of Excellence for Mathematics, University of Isfahan, Isfahan, Iran.Communicated by F. Giannessi     

Optimality Conditions for Vector Equilibrium Problems in Terms of Contingent Epiderivatives

In this article, we study some important properties of contingent epiderivatives concerning steady functions and a cone with a compact base along with its applications to establish necessary and sufficient optimality conditions for weakly efficient, Henig efficient, globally efficient and superefficient solutions for no constraints and constraints (it concludes cone constraint, equality constraint and a constraint set) vector equilibrium problems in terms of contingent epiderivatives. We also give some examples to illustrate obtained results.     

Improved ε-Constraint Method for?Multiobjective Programming

In this paper, we revisit one of the most important scalarization techniques used in multiobjective programming, the ε-constraint method. We summarize the method and point out some weaknesses, namely the lack of easy-to-check conditions for properly efficient solutions and the inflexibility of the constraints. We present two modifications that address these weaknesses by first including slack variables in the formulation and second elasticizing the constraints and including surplus variables. We prove results on (weakly, properly) efficient solutions. The improved ε-constraint method that we propose combines both modifications. The research of M. Ehrgott was partially supported by University of Auckland Grant 3602178/9275 and by Deutsche Forschungsgemeinschaft Grant Ka 477/27-1. The research of S. Ruzika was partially supported by Deutsche Forschungsgemeinschaft Grant HA 1795/7-2. The authors thank the anonymous referees, whose comments helped improving the presentation of the paper including a shorter proof of Theorem 3.1.     

Generalized (<Emphasis Type="Italic">F</Emphasis>,<Emphasis Type="Italic">ρ</Emphasis>)-Convexity and Duality in Nonsmooth Problems of Multiobjective Optimization

A mixed-type dual for a nonsmooth multiobjective optimization problem with inequality and equality constraints is formulated. We obtain weak and strong duality theorems for a mixed-type dual without requiring the regularity assumptions and the nonnegativeness of the Lagrange multipliers associated to the equality constraints. We apply also a nonsmooth constraint qualification for multiobjective programming to establish strong duality results. In this case, our constraint qualification assures the existence of positive Lagrange multipliers associated with the vector-valued objective function. This work was supported by Center of Excellence for Mathematics, University of Isfahan, Isfahan, Iran.     

Higher-Order Variational Sets and Higher-Order Optimality Conditions for Proper Efficiency in Set-Valued Nonsmooth Vector Optimization

Higher-order variational sets are proposed for set-valued mappings, which are shown to be more convenient than generalized derivatives in approximating mappings at a considered point. Both higher-order necessary and sufficient conditions for local Henig-proper efficiency, local strong Henig-proper efficiency and local λ-proper efficiency in set-valued nonsmooth vector optimization are established using these sets. The technique is simple and the results help to unify first and higher-order conditions. As consequences, recent existing results are derived. Examples are provided to show some advantages of our notions and results. This work was partially supported by the National Basic Research Program in Natural Sciences of Vietnam.     

Sufficiency and Duality in Multiobjective Variational Problems with Generalized Type I Functions

Recently Hachimi and Aghezzaf introduced the notion of (F,α,ρ,d)-type I functions, a new class of functions that unifies several concepts of generalized type I functions. Here, we extend the concepts of (F,α,ρ,d)-type I and generalized (F,α,ρ, d)-type I functions to the continuous case and we use these concepts to establish various sufficient optimality conditions and mixed duality results for multiobjective variational problems. Our results apparently generalize a fairly large number of sufficient optimality conditions and duality results previously obtained for multiobjective variational problems.     

On Optimality Conditions for Henig Efficiency and Superefficiency in Vector Equilibrium Problems

Abstract

Necessary optimality conditions for local Henig efficient and superefficient solutions of vector equilibrium problems involving equality, inequality, and set constraints in Banach space with locally Lipschitz functions are established under a suitable constraint qualification via the Michel–Penot subdifferentials. With assumptions on generalized convexity, necessary conditions for Henig efficiency and superefficiency become sufficient ones. Some applications to vector variational inequalities and vector optimization problems are given as well.     

Efficiency Conditions and Duality for a Class of Multiobjective Fractional Programming Problems

A class of constrained multiobjective fractional programming problems is considered from a viewpoint of the generalized convexity. Some basic concepts about the generalized convexity of functions, including a unified formulation of generalized convexity, are presented. Based upon the concept of the generalized convexity, efficiency conditions and duality for a class of multiobjective fractional programming problems are obtained. For three types of duals of the multiobjective fractional programming problem, the corresponding duality theorems are also established.     

Generalized hessian forC 1,1 functions in infinite dimensional normed spaces

The subject of this paper is the systematic study of second order notions concerning differentiable functions with Lipschitz derivative. The results and notions are motivated by recent papers of Cominetti, Correa and Hiriart-Urruty. The first goal of this paper is the comparison of several known second order directional derivatives. The second goal is the introduction of a generalized Hessian which is a set of certain symmetric bilinear forms. The relation of this generalized Hessian to other existing second order derivatives is also described. The research was supported by a grant from the National Science Foundation NSF-66-2270, which is gratefully acknowledged. Research supported by the Hungarian National Foundation for Scientific Research (OTKA), Grant No. T-016846 and by the Humboldt Foundation.     

Essential Components of the Set of Weakly Pareto-Nash Equilibrium Points for Multiobjective Generalized Games in Two Different Topological Spaces

In this paper, we study the existence and essential components of the set of weakly Pareto-Nash equilibrium points for multiobjective generalized games in two different uniform topological spaces. We obtain some new existence theorems. Examples show that the results are not identical in two different topological spaces.The author thanks two referees for careful reading of the paper and helpful comments.