Necessary optimality conditions and exact penalization for non-Lipschitz nonlinear programs

Abstract

When the objective function is not locally Lipschitz, constraint qualifications are no longer sufficient for Karush–Kuhn–Tucker (KKT) conditions to hold at a local minimizer, let alone ensuring an exact penalization. In this paper, we extend quasi-normality and relaxed constant positive linear dependence condition to allow the non-Lipschitzness of the objective function and show that they are sufficient for KKT conditions to be necessary for optimality. Moreover, we derive exact penalization results for the following two special cases. When the non-Lipschitz term in the objective function is the sum of a composite function of a separable lower semi-continuous function with a continuous function and an indicator function of a closed subset, we show that a local minimizer of our problem is also a local minimizer of an exact penalization problem under a local error bound condition for a restricted constraint region and a suitable assumption on the outer separable function. When the non-Lipschitz term is the sum of a continuous function and an indicator function of a closed subset, we also show that our problem admits an exact penalization under an extended quasi-normality involving the coderivative of the continuous function.