Summary: If the strengthened invex property holds for a constrained minimization problem, then a Karush-Kuhn-Tucker point is a strict minimum. The strict minimum property is preserved under small perturbations of the problem. This allows sufficient conditions to be given for a minimax, starting from Karush-Kuhn-Tucker conditions. They extend to vector-valued minimax and to nonsmooth (Lipschitz) problems. An example is provided to illustrate the strengthened invex property, also a discussion of quadratic-linear (nonconvex) programming implications.