## 變異型態的最小最大定理 VARIANT MINIMAX THEOREMS

2002

Chi-Nan Tsai
##### Abstract

The socalled minimax theorem means that if \$X\$ and \$Y\$ are two sets, and \$f\$ and \$g\$ are two real-valued functions defined on \$X imes Y\$, under some conditions the following inequality holds: \$\$inf_{yin Y}sup_{xin X}f(x,y)leq sup_{xin X}inf_{yin Y}g(x,y).\$\$ We will extend the two functions version of minimax theorems. Our purpose of this paper is three folds:\ (1)According to Lin and Yu's thesis: Two Functions Generalization of Horvath's Minimax Theorem, we will extend some theorems without convexity.\ (2)Without the condition of usual two functions version of minimax theorem: \$f\$ must be strictly lesser or equal to \$g\$, we replace it by a milder condition: \$\$sup_{xin X}f(x,y)leq sup_{xin X}g(x,y), orall yin Y.\$\$ However, we require some restrictions; such as, the functions \$f\$ and \$g\$ are {it jointly upward}, and their upper sets are connected.\ (3)Sometimes on some given region, the two functions version of minimax theorems is failure. By use of the properties of multifunctions, we define the {it X-quasiconcave} set, so that we can extend the two functions minimax theorem to the graph of the multifunction. In fact, we get the inequality: \$\$inf_{yin T(X)}sup_{xin T^{-1}(y)}f(x,y)leq sup_{xin X}inf_{yin T(x)}g(x,y),\$\$ where \$T\$ is a multifunction from \$X\$ to \$Y\$, and \${g}\$ is a {it X- quasiconcave} set of \$T\$.