Minimax Methods

Abstract

In many scientific and engineering applications it is often necessary to minimize the maximum of some quantity with respect to one or more independent variables. Algorithms that can be used to solve problems of this type are said to be minimax algorithms. In the case where the quantity of interest depends on a real-valued parameter w that belongs to a set S, the objective function can be represented by f(x, w) and the solution of the minimax problem pertaining to f(x, w) amounts to finding a vector variable x that minimizes the maximum of f(x, w) over w ∈ S. There is also a discrete version of this problem in which the continuous parameter w is sampled to obtain discrete values S = {w _i : i = 1, ..., L} ⊂ S and the corresponding minimax optimization problem is to find a vector x that minimizes the maximum of f(x, w _i) over w _i ∈ S _d .