Solving min-max problems and linear semi-infinite programs

For a min-max problem in the form of minx X maxt T {ft(x)}, the nondifferentiability of the max function F(x) ≡ maxt T {ft(x)} presents special difficulty in finding optimal solutions. We show that an entropic regularization procedure can provide a smooth approximation Fp(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 Fp(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.