We consider the problem [GRAPHICS] where F is an element of R-r, p is an element of R-m, and where c (.) and h (.) are C-1. Let phi(.) = Min(c(p,F)=0)h(p,F). We show, by means of simple examples, that phi(.) is, in general, discontinuous. We develop in this paper necessary conditions for the case where phi(.) is continuous ( but not necessarily differentiable). In an alternative approach ( which is computationally inferior), we treat the discontinuous case as well. We apply the results to robust control in linear systems where p stands for the ( structured) real parameter uncertainty vector and F stands for the control parameters vector. We demonstrate the results by means of examples.