对称向量拟均衡问题有效解的存在性

利用标量化方法建立对称向量拟均衡问题有效解的存在性定理。作为标量化方法的应用，利用这一方法得到向量变分不等式和拟向量变分不等式有效解的存在性定理。     

Newton-like methods for solving vector optimization problems

In the context of Euclidean spaces, we present an extension of the Newton-like method for solving vector optimization problems, with respect to the partial orders induced by a pointed, closed and convex cone with a nonempty interior. We study both exact and inexact versions of the Newton-like method. Under reasonable hypotheses, we prove stationarity of accumulation points of the sequences produced by Newton-like methods. Moreover, assuming strict cone-convexity of the objective map to the vector optimization problem, we establish convergence of the sequences to an efficient point whenever the initial point is in a compact level set.     

Lipschitz properties of the scalarization function and applications

The scalarization functions were used in vector optimization for a long period. Similar functions were introduced and used in economics under the name of shortage function or in mathematical finance under the name of (convex or coherent) measures of risk. The main aim of this article is to study Lipschitz continuity properties of such functions and to give some applications for deriving necessary optimality conditions for vector optimization problems using the Mordukhovich subdifferential.     

Necessary and sufficient optimality conditions for fractional multi-objective problems

Using a special scalarization, we give necessary optimality conditions for fractional multiobjective optimization problems. Under a generalized invexity, sufficient optimality conditions are also given. All over the article, the data are assumed to be continuous but not necessarily Lipschitz.     

SUPER EFFICIENCY AND ITS SCALARIZATION IN TOPOLOGICAL VECTOR SPACE

1. Introduction and PreliminariesRecently, Borwein and Zhuang[1,21 introduced the concept of super efficiency in normedlinear space. Super efficiency refines the notion of efficiency and other kinds of properefficiency; they provided concise scalar characterizations and duality results when the underlying decision problem is convex. They also established a Lagrange Multiplier Theoremfor super efficiency in convex settings and expressed super efficient points as saddle pointsof appropriate L…     

A necessary and sufficient condition for Pareto-optimal security strategies in multicriteria matrix games

In this paper, a scalar game is derived from a zero-sum multicriteria matrix game, and it is proved that the solution of the new game with strictly positive scalarization is a necessary and sufficient condition for a strategy to be a Pareto-optimal security strategy (POSS) for one of the players in the original game. This is done by proving that a certain set, which is the extension of the set of security level vectors in the criterion function space, is convex and polyhedral. It is also established that only a finite number of scalarizations are necessary to obtain all the POSS for a player. An example is included to illustrate the main steps in the proof.This work was done while the author was a Research Associate in the Department of Electrical Engineering at the Indian Institute of Science and was financially supported by the Council of Scientific and Industrial Research, Delhi, India.The author wishes to express his gratefulness to Professor U. R. Prasad for helpful discussions and to two anonymous referees for suggestions which led to an improved presentation.     

Gap Functions and Existence of Solutions to Generalized Vector Quasi-Equilibrium Problems

This paper deals with generalized vector quasi-equilibrium problems. By virtue of a nonlinear scalarization function, the gap functions for two classes of generalized vector quasi-equilibrium problems are obtained. Then, from an existence theorem for a generalized quasi-equilibrium problem and a minimax inequality, existence theorems for two classes of generalized vector quasi-equilibrium problems are established. This research is partially supported by the Postdoctoral Fellowship Scheme of The Hong Kong Polytechnic University and the National Natural Science Foundation of China.     

Approximate solutions and scalarization in set-valued optimization

Abstract

Certain notions of approximate weak efficient solutions are considered for a set-valued optimization problem based on vector and set criteria approaches. For approximate solutions based on the vector approach, a characterization is provided in terms of an extended Gerstewitz’s function. For the set approach case, two notions of approximate weak efficient solutions are introduced using a lower and an upper quasi order relations for sets and further compactness and stability aspects are discussed for these approximate solutions. Existence and scalarization using a generalized Gerstewitz’s function are also established for approximate solutions, based on the lower set order relation.     

Scalarization and Kuhn-Tucker-Like Conditions for Weak Vector Generalized Quasivariational Inequalities

We consider a weak vector generalized quasivariational inequality. By introducing a method of scalarization which does not require any assumption on the data and by using previous results of the authors concerning scalar generalized quasivariational inequalities, we present Kuhn-Tucker-like conditions for this problem in the case in which the set-valued operator of the constraints is defined by a finite number of inequalities     

Hölder Continuity of Solutions to Parametric Vector Equilibrium Problems with Nonlinear Scalarization

In this article, using the nonlinear scalarization approach by virtue of the nonlinear scalarization function, commonly known as the Gerstewitz function in the theory of vector optimization, Hölder continuity of solution mappings for both set-valued and single-valued cases to parametric vector equilibrium problems is studied. The nonlinear scalarization function is a powerful tool that plays a key role in the proofs, and its main properties (such as sublinearity, continuity, convexity) are fully employed. Especially, its locally and globally Lipschitz properties are provided and the Lipschitz property is first exploited to investigate the Hölder continuity of solutions.