Advanced Modeling and Optimization

Abstract for Paper 2 of Volume 1, Number 3, 1999, pp. 12-29


Convexity beyond vector spaces, alternative theorems and minimax equality


Anton Stefanescu
University of Bucharest, Faculty of Mathematics,
Academiei st., Bucharest 70109, Romania.
E­mail: anton@pro.math.unibuc.ro

Abstract

The concept of convexlike (concavelike) functions was introduced by Ky Fan (1953), who has proved the first minimax theorem without linear structure of the underlying spaces. Further extensions or generalizations of this concept have been used later in optimization and decision theory, but the most significant applications are in the framework of the game theory, where the strategy spaces are not endowed with natural algebraic structures. In the present paper one introduces new convexity and connectedness conditions and establishes the relationships with other known convexlike type properties. The main results concern the minimax equality in a topological framework. They generalize classical minimax theorems of Fan and KĻonig, and are independent of most similar results known in the literature. The proofs make use of some special alternative theorems which also hold in a pure topological framework, without any vector space structure. Key Words: minimax equality, theorem of the alternative, convexlike, connectedness.