TWO – FUNCTION EXTENSIONS OF SOME MINIMAX THEOREMS OF RICCERI

In recent years the minimax theorem of John von Neumann has found numerous new extensions, due to Irle [1985], Kindler [1990], Simons [1990][1991] and others, with the aim to remove from the assumptions the last remnants of linear and convex structures, and to install assumptions of more comprehensive kinds instead. The talk wants to present an extension and unification, due to the author in cooperation with Frank Zartmann, of the recent main contributions. The principal results are a quantitative theorem in the spirit of the concave-convexlike minimax theorem of Ky Fan [1953], and a topological theorem in the spirit of the quasiconcave-convex minimax theorem of Sion [1958]. A further main contribution is to decompose the minimax relation into independent halfs, such that the minimax theorems quoted above — and hence the bulk of the minimax theorems known so far — appear as unions of one-sided theorems, which then can be combined at will to minimax theorems of mixed types in the spirit of Terkelsen [1972].

Preface. Nonlinear Two Functions Minimax Theorems Cao-Zong Cheng, Bor-Luh Lin. Weakly Upward-Downward Minimax Theorem Cao-Zong Cheng, et al. A Two-Function Minimax Theorem A. Chinni. Generalized Fixed-Points and Systems of Minimax Inequalities P. Deguire. A Minimax Inequality for Marginally Semicontinuous Functions G.H. Greco, M.P. Moschen. On Variational Minimax Problems under Relaxed Coercivity Assumptions J. Gwinner. A Topological Investigation of the Finite Intersection Property C.D. Horvath. Minimax Results and Randomization for Certain Stochastic Games A. Irle. Intersection Theorems, Minimax Theorems and Abstract Connectedness J. Kindler. K-K-M-S Type Theorems in Infinite Dimensional Spaces H. Komiya. Hahn-Banach Theorems for Convex Functions M. Lassonde. Two Functions Generalization of Horvath's Minimax Theorem Bor-Luh Lin, Feng-Shuo Yu. Some Remarks on a Minimax Formulation of a Variational Inequality G. Mastroeni. Network Analysis M.M. Neumann, M.V. Velasco. On a Topological Minimax Theorem and its Applications B. Ricceri. Three Lectures on Minimax and Monotonicity S. Simons. Fan's Existence Theorem for Inequalities Concerning Convex Functions and its Applications W. Takahashi. An Algorithm for the Multi-Access Channel Problem Peng-Jung Wan, et al. Author Index.

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.

A new topological minimax theorem is established for functions on $${C\times \mathbb R}$$ where C is a topological space. Although this theorem includes as special cases most important recent results on this subject, its proof is surprisingly simple. An application to nonlinear optimization theory is considered.

does hold. The relevant literature is by now really impressive. For a first approach to it, we refer the reader to the references quoted in [4], [5], [7], [9]. However, as well stressed by the lucid Introduction of [9], it is rather easy to recognize a few research currents within which practically each existing contribution can be located. In particular, one of these currents is that concerning the so called topological mini-max theorems, the ancestors of which are the results of [13] and, more properly, of [16] (see also [2]). For a sharp discussion on such kind 0f theorems, we refer to a very recent paper by H. K6nig [8] which also contains the best results in this area, up to now. Since the aim of the present paper is to establish some new topological mini-max theorems as applications of an alternative principle for multifunctions (several further consequences of which will be presented in successive papers), we consider just [8] as a starting point to introduce the main novelties of our results. So, we state now a particular case of the main theorem of [8].

Two-Function Topological Minimax Theorems

Topological Minimax Theorems and Approximation

A Topological KKM Theorem and Minimax Theorems

In this paper, we introduce sufficient conditions for the non—empty intersection of two set—valued mappings in topological spaces. As applications, some topological minimax inequalities for two functions in which one of them is separately lower (or upper) semicontinuous are given. Finally, by employing our topological intersection theorems for two set—valued mappings, some other minimax inequalities have been derived without separately lower (or upper) sernicontinuity under but with another condition. These results are topological versions of corresponding minimax inequalities for two functions due to Fan (1964) and Sion (1958) in topological vector spaces

Some noncompact topological and mixed minimax theorems involving compactly locally upward and finitely weakly downward functions are proved.

In this paper, a topological intersection theorem of a family of sets with H-convex sections is established. As an application to this theorem, the Nash equilibrium of abstract economy, the Ky-Fan minimax theorem, and an existence problem of solution for quasi-variational inequality are studied. The results presented in this paper modify and extend the corresponding results given in the literature.

A noncompact topological minimax theorem

Let X, Y be two non-empty sets and let f be a real function defined on X × Y. The classical minimax problem is to find suitable conditions under which the equality $$ \mathop {\sup }\limits_{y \in Y} \,\mathop {\inf }\limits_{x \in X} \,f\left( {x,y} \right) = \mathop {\inf }\limits_{x \in X} \,\mathop {\sup }\limits_{y \in Y} \,f\left( {x,y} \right) $$ ((1.1)) does hold.KeywordsLower SemicontinuousTopological Vector SpaceReal Banach SpaceReal Vector SpaceMinimax TheoremThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A two functions, noncompact topological minimax theorem

On the basis of a new topological minimax theorem, a simple and unified approach is developed to Lagrange duality in nonconvex quadratic programming. Diverse generalizations as well as equivalent forms of the S-Lemma, providing a thorough study of duality for single constrained quadratic optimization, are derived along with new strong duality conditions for multiple constrained quadratic optimization. The results allow many quadratic programs to be solved by solving one or just a few SDP’s (semidefinite programs) of about the same size, rather than solving a sequence, often infinite, of SDP’s or linear programs of a very large size as in most existing methods.