Constrained Minima and Lipschitzian Penalties in Metric Spaces

It is well known that a local minimizer of a constrained optimization problem with Lipschitzian objective is a free local minimizer of an assigned penalty function if the constraints satisfy an appropriate regularity condition. We use an upper Lipschitz property (L1) as regularity concept and present locally Lipschitz penalty functions defined on the whole space for arbitrary constraint maps of this type. We give conditions under which the maximum of the penalties of finitely many multifunctions is a valid penalty function for the intersection of these multifunctions. Further, the same statements will be derived under other regularity assumptions, namely, for calm or pseudo-Lipschitz constraints which violate (L1), by showing that some submapping of a calm map always has property (L1) and possesses (locally) the same penalties. In this way our penalizations induce in a unified manner, via known properties of free local minimizers for Lipschitz functions only, primal and dual necessary conditions for these basic notions of regularity.