Mixed min-max optimization problems

We consider a min-max optimization using relaxed controls. Such problems have been considered using Dubovitskii and Milyutin Theory and convex analysis [3]. We reformulate the problem where a unilateral constraint is introduced. We deal with the reformulated problem using penalization and obtain additional optimality conditions than presented in [3]. In the special case where the controls appear linearly and the control sets are convex polyhedra the relaxed controls are ordinary controls taking values in the vertices of the polyhedra.