On minimization subject to cone constraints

Abstract

Modified Lagrangian conditions, which are necessary for a constrained minimum without requiring any constraint qualification, are obtained for an arbitrary differentiable objective function and pseudoconvex constraint functions. The results apply to constraints with closed convex cones, and to spaces of any dimensions. The necessary conditions become also sufficient with pseudoconvex objective and quasiconvex constraint functions. They are applied to semi-infinite programming, lexicographic optimization, and optimal control with state constraints.