Characterizations of the Benson Proper Efficiency for Nonconvex Vector Optimization

Under generalized cone-subconvexlikeness for vector-valued mappings in locally-convex Hausdorff topological vector spaces, a Gordan-form alternative theorem is derived. Some characterizations of the Benson proper efficiency under this generalized convexity are established in terms of scalarization, Lagrangian multipliers, saddle-point criterion, and duality.     

Generalized Motzkin Theorems of the Alternative and Vector Optimization Problems

In this paper, we introduce a definition of generalized convexlike functions (preconvexlike functions). Then, under the weakened convexity, we study vector optimization problems in Hausdorff topological linear spaces. We establish some generalized Motzkin theorems of the alternative. By use of these theorems of the alternative, we obtain some Lagrangian multiplier theorems. A saddle-point theorem and a scalarization theorem are also derived.Communicated by F. GiannessiThe author thank Ginndomenico Mastrocni for helpful and useful comments.     

Theorems of the alternative and optimality conditions

A theorem of the alternative is stated for generalized systems. It is shown how to deduce, from such a theorem, known optimality conditions like saddle-point conditions, regularity conditions, known theorems of the alternative, and new ones. Exterior and interior penalty approaches, weak and strong duality are viewed as weak and strong alternative, respectively.     

On saddle-point optimality in differential games

A family of two-person, zero-sum differential games in which the admissible strategies are Borel measurable is defined, and two types of saddle-point conditions are introduced as optimality criteria. In one, saddle-point candidates are compared at each point of the state space with all playable pairs at that point; and, in the other, they are compared only with strategy pairs playable on the entire state space. As a theorem, these two types of optimality are shown to be equivalent for the defined family of games. Also, it is shown that a certain closure property is sufficient for this equivalence. A game having admissible strategies everywhere constant, in which the two types of saddle-point candidates are not equivalent, is discussed.This paper is based on research supported by ONR.     

Duality theory in multiobjective programming

In this paper, a multiobjective programming problem is considered as that of finding the set of all nondominated solutions with respect to the given domination cone. Two point-to-set maps, the primal map and the dual map, and the vector-valued Lagrangian function are defined, corresponding to the case of a scalar optimization problem. The Lagrange multiplier theorem, the saddle-point theorem, and the duality theorem are derived by using the properties of these maps under adequate convexity assumptions and regularity conditions.     

General KKM Theorem with Applications to Minimax and Variational Inequalities

In this paper, a general version of the KKM theorem is derived by using the concept of generalized KKM mappings introduced by Chang and Zhang. By employing our general KKM theorem, we obtain a general minimax inequality which contains several existing ones as special cases. As applications of our general minimax inequality, we derive an existence result for saddle-point problems under general setting. We also establish several existence results for generalized variation inequalities and generalized quasi-variational inequalities.     

Generalized Equilibrium Problems and Generalized Complementarity Problems

From a general minimax inequality or an abstract lopsided saddle-point theorem, we deduce general Karamardian-type equilibrium theorems and generalized complementarity theorems. Our new results extend a number of well-known earlier works of many authors.     

On the sufficiency of the linear maximum principle for discrete-time control problems

This note presents a family of linear maximum principles for the discrete-time optimal control problem, derived from the saddle-point theorem of mathematical programming. Some simple examples illustrate the applicability of the main theoretical results.     

Equivalence between a generalized fenchel duality theorem and a saddle-point theorem for fractional programs

The convex case of a fractional program is considered. By the aid of a duality theory for mathematical programming involving a maximum and minimum operator, and by defining new operators, we obtain three equivalent duality theorems for three pairs of primal-dual programs. The second and the third one are the saddle-point theorem and a generalized Fenchel duality theorem, respectively.     

BSDEs with random default time and related zero-sum stochastic differential games

In this Note we are concerned with backward stochastic differential equations with random default time. The equations are driven by Brownian motion as well as a mutually independent martingale appearing in a defaultable setting. We show that these equations have unique solutions and a comparison theorem for their solutions. As an application, we get a saddle-point strategy for the related zero-sum stochastic differential game problem.     

On duality theory in multiobjective programming

In this paper, we study different vector-valued Lagrangian functions and we develop a duality theory based upon these functions for nonlinear multiobjective programming problems. The saddle-point theorem and the duality theorem are derived for these problems under appropriate convexity assumptions. We also give some relationships between multiobjective optimizations and scalarized problems. A duality theory obtained by using the concept of vector-valued conjugate functions is discussed.The author is grateful to the reviewer for many valuable comments and helpful suggestions.     

A Topological Minimax Theorem

We present a topological minimax theorem (Theorem 2.2). The topological assumptions on the spaces involved are somewhat weaker than those usually found in the literature. Even when reinterpreted in the convex setting of topological vector spaces, our theorem yields nonnegligible improvements, for example, of the Passy–Prisman theorem and consequently of the Sion theorem, contrary to most results on topological minimax. This work is part of our ongoing effort to elaborate a coherent theory of minimax.     

非光滑非凸向量极值问题的真有效解

本文考虑非光滑非凸向量极值问题的真有效解,其主要结果如下:(1)Borwein真有效解与Benson真有效解的等价性;(2)向量极值问题的真有效解与标量极值问题的最优解的等价性;(3)广义鞍点定理;(4)真有效解的必要和充分条件。     

The generalized optimality conditions of multiobjective programming problem in topological vector space

In this study, an alternative theorem for the subconvexlike mapping in topological vector space is established. With this alternative theorem as an aid, the generalized Fritz John conditions and the generalized Kuhn-Tucker conditions in terms of Gâteaux derivatives of multiobjective programming problem in the ordered topological vector space are given.     

Constrained markov decision processes with compact state and action spaces: the average case

Constrained Markov decision processes with compact state and action spaces are studied under long-run average reward or cost criteria. By introducing a corresponding Lagrange function, a saddle-point theorem is given, by which the existence of a constrained optimal pair of initial state distribution and policy is shown. Also, under the hypothesis of Doeblin, the functional characterization of a constrained optimal policy is obtained     

Generalized type theorems in topological spaces with applications

In this work, some new generalized type theorems for generalized mappings with compactly open values are established in topological spaces without any convexity assumptions. As applications, a Ky Fan type matching theorem, fixed point theorem and coincidence theorem are obtained in topological spaces. These results generalize some known results from the recent literature.     

拓扑线性空间中的Drop定理和Phelps引理以及Ekeland变分原理

将Phelps引理, Ekeland变分原理, Pareto有效性定理推广到拓扑线性空间,同时证明了这三个定理与郑喜印证明的拓扑线性空间中的Drop定理彼此等价．     

广义概率A—proper 映象的拓扑度及其不动点定理

本文建立了M—PN 空间中广义概率A-proper 映象的拓扑度及其不动点定理。它是广义A—proper 映象及其不动点定理的推广。     

线性拓扑空间中标量化定理的一个证明

本文对标量化问题的一个重要定理 [1]在线性拓扑空间中给予了证明 ,从而使该定理应用到线性拓扑空间 .     

Coincidence Theorems with Applications to Minimax Inequalities, Section Theorem, Best Approximation and Multiobjective Games in Topological Spaces

Some new coincidence theorems involving admissible set-valued mappings are proved in general noncompact topological spaces. As applications, some new minimax inequalities, section theorem, best approximation theorem, existence theorems of weighted Nash equilibria and Pareto equilibria for multiobjective games are given in general topological spaces.