An algorithm for generalized semi-infinite min-max problems using exact penalties

We develop an implementable algorithm for the solution of a class of generalized semi-infinite min-max problems. First, we use exact penalties to convert a generalized semi-infinite min-max problem into an infinite family of semi-infinite min-max-min problems. Next, the inner min-function is smoothed and the semi-infinite max part is approximated, using discretization, to obtain a three-parameter family of finite min-max problems. We show that the min-max-min problems have identical solutions to those of the original problem when the penalty is sufficiently large and that the solutions of the finite min-max problems converge to solutions of the original problem when the smoothing and discretization parameters go to infinity, provided the penalty parameter is sufficiently large. A numerical example demonstrates the viability of the algorithm.