Implementing a smooth exact penalty function for equality-constrained nonlinear optimization

TL;DR

This paper introduces an efficient implementation of a smooth exact penalty function for equality-constrained nonlinear optimization, demonstrating its practicality and potential advantages over traditional methods, especially in PDE-constrained problems.

Contribution

It develops a computationally feasible approach to implement Fletcher's smooth penalty function, including strategies for solving the linear systems efficiently and adapting to factorization-free algorithms.

Findings

The proposed method is comparable in cost to existing nonlinear optimization techniques.

Efficient solutions are possible using single factorization storage per iteration.

The approach performs well on standard test problems and PDE-constrained optimization cases.

Abstract

We develop a general equality-constrained nonlinear optimization algorithm based on a smooth penalty function proposed by Fletcher (1970). Although it was historically considered to be computationally prohibitive in practice, we demonstrate that the computational kernels required are no more expensive than other widely accepted methods for nonlinear optimization. The main kernel required to evaluate the penalty function and its derivatives is solving a structured linear system. We show how to solve this system efficiently by storing a single factorization each iteration when the matrices are available explicitly. We further show how to adapt the penalty function to the class of factorization-free algorithms by solving the linear system iteratively. The penalty function therefore has promise when the linear system can be solved efficiently, e.g., for PDE-constrained optimization problems where efficient preconditioners exist. We discuss extensions including handling simple constraints explicitly, regularizing the penalty function, and inexact evaluation of the penalty function and its gradients. We demonstrate the merits of the approach and its various features on some nonlinear programs from a standard test set, and some PDE-constrained optimization problems.