Solving min-max problems and linear semi-infinite programs

For a min-max problem in the form of min(x is an element of X)max(t is an element of T) {ft(x)}, the non differentiability of the max function F(x) = max(t is an element of T) {ft(x)} presents special difficulty in finding optimal solutions. We show that an entropic regularization procedure can provide a smooth approximation F-p(x) that uniformly converges to F(x) over X, as p tends to infinity. In this way, with p being sufficiently large, minimizing the smooth function F-p(x) over X provides a very accurate approximate solution to the min-max problem. When this approach is applied to solve linear semi-infinite programming problems, the previously proposed ''unconstrained convex programming approach'' is shown to be a special case.