Implementing a smooth exact penalty function for general constrained nonlinear optimization

TL;DR

This paper introduces a practical implementation of Fletcher's smooth penalty function for constrained nonlinear optimization, demonstrating its efficiency and applicability to PDE-constrained problems with promising numerical results.

Contribution

It develops a computationally feasible algorithm based on Fletcher's smooth penalty function, with new strategies for solving linear systems efficiently in nonlinear optimization.

Findings

Comparable computational cost to existing methods

Effective for PDE-constrained optimization with preconditioning

Successful numerical results on test problems

Abstract

We build upon Estrin et al. (2019) to develop a general constrained nonlinear optimization algorithm based on a smooth penalty function proposed by Fletcher (1970, 1973b). Although Fletcher's approach has historically been considered impractical, we show that the computational kernels required are no more expensive than those in other widely accepted methods for nonlinear optimization. The main kernel for evaluating the penalty function and its derivatives solves structured linear systems. When the matrices are available explicitly, we store a single factorization each iteration. Otherwise, we obtain a factorization-free optimization algorithm by solving each linear system iteratively. The penalty function shows promise in cases where the linear systems can be solved efficiently, e.g., PDE-constrained optimization problems when efficient preconditioners exist. We demonstrate the merits of the approach, and give numerical results on several PDE-constrained and standard test problems.