On the entropic regularization method for solving min-max problems with applications

Consider a min-max problem in the form of min(x is an element of max1 less than or equal to i less than or equal to m){f(i)(x)}. It is well-known that the non-differentiability of the max function F(x) equivalent to max(1 less than or equal to i less than or equal to m){f(i)(x)} presents difficulty in finding an optimal solution. An entropic regularization procedure provides a smooth approximation F-p(x) that uniformly converges to F(x) over X with a difference bounded by ln(m)/p, for p > 0. In this way, with p being sufficiently large, minimizing the smooth function F,(x) over X provides a very accurate solution to the min-max problem. The same procedure can be applied to solve systems of inequalities, linear programming problems, and constrained min-max problems.