變異型態的最小最大定理

Abstract

所謂兩個函數的最小最大定理(minimax theorem) , 是指在給定的兩個集合$X$ 和 $Y$ 中 , 研究定義在 $X imes Y$ 上的兩個實值函數 $f$ 和 $g$ , 是否可以得到下列不等 式 $$inf_{yin Y}sup_{xin X}f(x,y)leq sup_{xin X}inf_{yin Y}g(x,y).$$ 此篇論文將做進一步推廣 , 主要的推論有三層 :\ (1) 根據Lin和Yu的研究論文 : Two Functions Generalization of Horvath's Minimax Theorem, 我們將推廣出一些不需要凸性的最小最大定理. vspace{1cm} (2) 打破一般關於兩個函數的最小最大定理中所規定 $f$ 必須嚴格小於或等於 $g$ 的條 件 , 取而代之的是 $$sup_{xin X}f(x,y)leq sup_{xin X}g(x,y), orall yin Y.$$ 當然 , 其中的兩個函數需稍作限制 , 包括 : 兩個函數形成聯合向上($jointly\nupward$) 函數關係 , 以及它們所形成的上集合($upper set$)必需為連通的$dots$ 等. vspace{1cm} (3) 有時候在某個定義域上兩個函數的最小最大定理不會成立 , 但是若在此時稍微限制定義域的範圍後 , 最小最大定理便可以成立了 ! 於是我們利用了多值函數的一些性質 , 定義$X$- 擬凹集合 , 推廣出在多值函數上的最小最大定理 , 而得到下列變異型態的最小最大不等式 $$inf_{yin T(X)}sup_{xin T^{-1}(y)}f(x,y)leq sup_{xin X}inf_{yin T(x)}g(x,y).$$ 其中 , $T$ 為由 $X$ 對應到 $Y$ 的多值函數 , ${g}$ 則是相應於 $T$ 的$X$- 擬凹集合.

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 $T$.