A simple derivation of necessary conditions for static minmax problems

Abstract

where +(., *): Rn x Rm -+ R is Cl. Y is a specified subset of Rm and X = {x 1 C(x) < 01. The function C(e): Rn R’ is Cl on Ii”. Necessary conditions which the minimizing point must satisfy are given in [l-3] in terms of the directional derivative off(*). This results in a Lagrange multiplier rule which is an inequality. In [4], the necessary conditions are in the form of a Lagrange multiplier rule that is an equality. Here, we derive the results of [4] in an alternate and simpler fashion. Our approach is to replace the above minmax problem, as suggested in [S], by a related nonlinear programming problem where the subsidiary conditions consist of an infinite number of inequalities. The resulting programming problem is called the Fritz John problem [6, 71. A relationship between the solution of the Fritz John problem and the minmax problem is presented. This relationship, in conjunction with the necessary conditions for the Fritz John problem, lead directly to necessary conditions for the minmax problem which are identical to those in [4]. We shall assume that Y is compact. Then for every 4 E X there exists a 9 E Y with the property that 4(&y) = supyey +(f, y) and we can replace “sup” by “max”. The vector of partial derivatives @(x,y)/& will be denoted by +r(x, y). Similarly, C,,(x) = aC,(x)/a.r.