Smooth Transformation of the Generalized Minimax Problem

We consider the generalized minimax problem, that is, the problem of minimizing a function (x)=F(g ₁(x),...,g _m(x)), where F is a smooth function and each g _i is the maximum of a finite number of smooth functions. We prove that, under suitable assumptions, it is possible to construct a continuously differentiable exact barrier function, whose minimizers yield the minimizers of the function . In this way, the nonsmooth original problem can be solved by usual minimization techniques for unconstrained differentiable functions.