Liang-Ju Chu, and Chi-Nan Tsai. "Minimax theorems without changeless proportion." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 23.1 (2003): 55-92. <http://eudml.org/doc/271509>.
@article{Liang2003,
abstract = {The so-called minimax theorem means that if X and Y are two sets, and f and g are two real-valued functions defined on X×Y, then under some conditions the following inequality holds:
$inf_\{y∈Y\} sup_\{x∈X\} f(x,y) ≤ sup_\{x∈X\} inf_\{y∈Y\} g(x,y)$.
We will extend the two functions version of minimax theorems without the usual condition: f ≤ g. We replace it by a milder condition:
$sup_\{x∈X\} f(x,y) ≤sup_\{x∈X\}g(x,y)$, ∀y ∈ Y.
However, we require some restrictions; such as, the functions f and g are jointly upward, and their upper sets are connected. On the other hand, by using some properties of multifunctions, we define X-quasiconcave sets, so that we can extend the two functions minimax theorem to the graph of the multifunction. In fact, we get the inequality:
$inf_\{y∈T(X)\} sup_\{x∈T^\{-1\}(y)\} f(x,y) ≤ sup_\{x∈X\} inf_\{y∈T(x)\} g(x,y)$,
where T is a multifunction from X to Y.},
author = {Liang-Ju Chu, Chi-Nan Tsai},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {minimax theorems; t-convex functions; upward functions; jointly upward functions; X-quasiconcave sets; -convex functions; quasiconvex sets; multifunctions},
language = {eng},
number = {1},
pages = {55-92},
title = {Minimax theorems without changeless proportion},
url = {http://eudml.org/doc/271509},
volume = {23},
year = {2003},
}
TY - JOUR
AU - Liang-Ju Chu
AU - Chi-Nan Tsai
TI - Minimax theorems without changeless proportion
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2003
VL - 23
IS - 1
SP - 55
EP - 92
AB - The so-called minimax theorem means that if X and Y are two sets, and f and g are two real-valued functions defined on X×Y, then under some conditions the following inequality holds:
$inf_{y∈Y} sup_{x∈X} f(x,y) ≤ sup_{x∈X} inf_{y∈Y} g(x,y)$.
We will extend the two functions version of minimax theorems without the usual condition: f ≤ g. We replace it by a milder condition:
$sup_{x∈X} f(x,y) ≤sup_{x∈X}g(x,y)$, ∀y ∈ Y.
However, we require some restrictions; such as, the functions f and g are jointly upward, and their upper sets are connected. On the other hand, by using some properties of multifunctions, we define X-quasiconcave sets, so that we can extend the two functions minimax theorem to the graph of the multifunction. In fact, we get the inequality:
$inf_{y∈T(X)} sup_{x∈T^{-1}(y)} f(x,y) ≤ sup_{x∈X} inf_{y∈T(x)} g(x,y)$,
where T is a multifunction from X to Y.
LA - eng
KW - minimax theorems; t-convex functions; upward functions; jointly upward functions; X-quasiconcave sets; -convex functions; quasiconvex sets; multifunctions
UR - http://eudml.org/doc/271509
ER -
