Co,,,p B Ma,i,\ xrrlr A,>,h zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Vd II. Irio 12. pp zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 1165-116’). zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA IYXS Prmcd ,n Circa~ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Hr,la,n zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA SOLUTION OF MINIMAX PROBLEMS USING EQUIVALENT DIFFERENTIABLE FUNCTIONS zyxwvutsrqponmlkjihgfedcb REUVEN CHEN-~ School of Physica and Astronomy. Raymond and Beverly Sackler Faculty of Exact Sciences. Tel-Aviv University, Tel-Aviv. Israel (Received June 1985) Communicated by I. N. Katz A bstract-A method is proposed for the solution of minimax optimization problems in which the individual functions involved are convex. The method consists of solving a problem with an objective function which ia the sum of high powers or strong exponentials of the separate components of the original objective function. The resulting objective function. which is equivalent at the limit to the minimax one. is shown to be smooth as well as convex. Any efficient nonlinear programming method can be utilized for solving the equivalent problem. A number of examples are discussed. I. INTRODUCTION The minimax optimization problem is that of finding min {max [f,(x)ll, x 111S11 (1) where x = (x,, . xk). The functions f;(x) are assumed to be smooth, however, the main difficulty in solving (1) is usually related to the kinks in the objective function F(x) = max Lfi(x)l ,S!S,! These kinks are points at which F(x) is not differentiable and in most cases, the solution point occurs at such a kink. The theory of nonlinear minimax has been thoroughly studied by Dem’yanov and Malozemov[9] who investigated the differentiability of the maximum function, and discussed the necessary and sufficient conditions for local and global solutions as well as the properties of the maximum problem. More recent theoretical work has been given by Ben-Tal and Zowe[3] and by Drezner[ lo]. The importance of the nonlinear minimax problem seems to exceed sub- stantially the scope of solving problems which are initially of this nature since as shown by Bandler and Charalambous[2]. any nonlinear programming problems with nonlinear constraints can be transformed into an equivalent unconstrained minimax problem. The numerical methods utilized for solving nonlinear minimax problems consisted mainly of approximating the F(x) = max,,,,,, [f,(x)] function by close enough functions in which the kinks are smoothed out. Charalambous and Bandler[4] suggested the use of as long as f,(x) are all positive and zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA P is a large enough positive number. The result is a smooth approximation for max,,,,,, [f,(x)]. Intuitively, if f;(x) are all positive valued, raising each of them to a high power will emphasize the largest one as compared to the others so that if fN(x) > f,(x) V i # N. then *Work done while in residence in the Institute for Mathematics and its Applications. Umversity of Minnesota. Mmneapohs. MN 55455 C.S.A. 1165