We consider global optimization problems of a minmax function subject to constraints with quantifiers. The goal is to minimize over x the maximum over y of f(x,y), and constraints with quantifiers must be satisfied for a set of value (for all y in S). We present an algorithm to address this non-smooth objective function and the constraints with quantifiers in the same way. Our approach is a set-valued method based on two nested branch-and-bound algorithm, using interval analysis. This problem arises in structured robust control, semi-infinite optimization or minmax optimization.