A Minimax Theorem forL-0-Valued Functions on Random Normed Modules

Shien Zhao,Yuan Zhao

doi:10.1155/2013/704251

Abstract

We generalize the well-known minimax theorems toL¯0-valued functions on random normed modules. We first give some basic properties of anL0-valued lower semicontinuous function on a random normed module under the two kinds of topologies, namely, the (ε,λ)-topology and the locallyL0-convex topology. Then, we introduce the definition of random saddle points. Conditions for anL0-valued function to have a random saddle point are given. The most greatest difference between our results and the classical minimax theorems is that we have to overcome the difficulty resulted from the lack of the condition of compactness. Finally, we, using relations between the two kinds of topologies, establish the minimax theorem ofL¯0-valued functions in the framework of random normed modules and random conjugate spaces.

Highlights

The classical minimax theorem, which originated from game theory, is an important content of nonlinear analysis
The most greatest difference between our results and the classical minimax theorems is that we have to overcome the difficulty resulted from the lack of the condition of compactness
The remainder of this paper is organized as follows: in Section 2 we will briefly collect some necessary known facts; in Section 3 we will give some basic properties of an L0-valued lower semicontinuous function on a random normed module under the two kinds of topologies, namely, Theorems 26 and 28; in Section 4 we will present the definition of random saddle points and prove our main result

Summary

Introduction

The classical minimax theorem, which originated from game theory, is an important content of nonlinear analysis. In [15] Guo et al establish a complete random convex analysis over RN modules and RLC modules by simultaneously considering the two kinds of topologies in order to provide a solid analytic foundation for the module approach to conditional risk measures. The remainder of this paper is organized as follows: in Section 2 we will briefly collect some necessary known facts; in Section 3 we will give some basic properties of an L0-valued lower semicontinuous function on a random normed module under the two kinds of topologies, namely, Theorems 26 and 28; in Section 4 we will present the definition of random saddle points and prove our main result

Preliminaries

Some Basic Properties of L0-Valued Lower Semicontinuous Functions

Main Results