Smooth transformation of the generalized minimax problem

We consider the generalized minimax problem, that is, the problem of minimizing a function phi (x) = F(g(l)(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 phi. In this way, the nonsmooth original problem can be solved by usual minimization techniques for unconstrained differentiable functions.