# Necessary Optimality Conditions and Exact Penalization for Non-Lipschitz Nonlinear Programs

## TL;DR

This paper extends optimality conditions and exact penalization results to non-Lipschitz nonlinear programs, broadening the theoretical understanding and applicability of KKT conditions and penalty methods in such challenging settings.

## Contribution

It introduces generalized constraint qualifications and establishes exact penalization results for non-Lipschitz objectives, expanding the scope of optimality theory.

## Key findings

- Extended quasi-normality and RCPLD conditions ensure KKT necessity for non-Lipschitz problems.
- Derived exact penalization results for problems with composite and indicator functions.
- Provided conditions under which local minimizers coincide with penalized problem solutions.

## 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 (RCPLD) 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.