A unifying theory of exactness of linear penalty functions II: parametric penalty functions

TL;DR

This paper develops a comprehensive theory for exact parametric penalty functions in constrained optimization, highlighting their smoothness, conditions for exactness, and convergence, unifying and improving existing results.

Contribution

It introduces a general framework for exact parametric penalty functions, providing necessary and sufficient conditions, convergence analysis, and applications to existing penalty methods.

Findings

Parametric penalty functions can be both smooth and exact.

Necessary and sufficient conditions for exactness are established.

Convergence results for the penalty method are proved.

Abstract

In this article we develop a general theory of exact parametric penalty functions for constrained optimization problems. The main advantage of the method of parametric penalty functions is the fact that a parametric penalty function can be both smooth and exact unlike the standard (i.e. non-parametric) exact penalty functions that are always nonsmooth. We obtain several necessary and/or sufficient conditions for the exactness of parametric penalty functions, and for the zero duality gap property to hold true for these functions. We also prove some convergence results for the method of parametric penalty functions, and derive necessary and sufficient conditions for a parametric penalty function to not have any stationary points outside the set of feasible points of the constrained optimization problem under consideration. In the second part of the paper, we apply the general theory of exact parametric penalty functions to a class of parametric penalty functions introduced by Huyer and Neumaier, and to smoothing approximations of nonsmooth exact penalty functions. The general approach adopted in this article allowed us to unify and significantly sharpen many existing results on parametric penalty functions.