Basics of Minimax Algorithms

Abstract

Minimax problems are a very important class of nonsmooth optimization problems. They occur in curve fitting, engineering design, optimal control and many other situations (see [26]) for some specific examples). They are also among the best understood nonsmooth optimization problems, particularly when they involve maxima of smooth functions. There is now a considerable literature dealing with minimax problems and we present a selected list of publications in our references section (see [2], [4], [6], [7], [8], [9], [11], [12], [18], [20], [21], [22], [23], [24], [25], [26], [28], [29], [31], [32]). Looking over these papers, the reader will find that several approaches to minimax algorithms are possible, some of which yield first order methods, while others yield superlinearly converging ones. In this chapter we examine a particularly simple approach to the construction of minimax algorithms, which yields first order methods only.