Smooth exact penalty functions II: a reduction to standard exact penalty functions

TL;DR

This paper establishes a theoretical connection between a new smooth exact penalty function and the standard nonsmooth penalty function in constrained optimization, providing insights into their equivalence and parameter behavior.

Contribution

It proves the equivalence of exactness between the new smooth penalty function and the standard nonsmooth penalty, and analyzes the asymptotic behavior of the penalty parameters.

Findings

Smooth penalty function is exact iff the standard penalty is exact.

The smooth penalty parameter asymptotically behaves as the square of the standard penalty parameter.

Reducing the penalty parameter can be achieved through a simple method.

Abstract

A new class of smooth exact penalty functions was recently introduced by Huyer and Neumaier. In this paper, we prove that the new smooth penalty function for a constrained optimization problem is exact if and only if the standard nonsmooth penalty function for this problem is exact. We also provide some estimates of the exact penalty parameter of the smooth penalty function, and, in particular, show that it asymptotically behaves as the square of the exact penalty parameter of the standard $\ell_1$ penalty function. We briefly discuss a simple way to reduce the exact penalty parameter of the smooth penalty function, and study the effect of nonlinear terms on the exactness of this function.