IMPLEMENTING A SMOOTH EXACT PENALTY FUNCTION FOR EQUALITY-CONSTRAINED NONLINEAR OPTIMIZATION

作     者：Estrin, Ron Friedlander, Michael P. Orban, Dominique Saunders, Michael A. 

作者机构：Stanford Univ Inst Computat & Math Engn Stanford CA 94305 USA Univ British Columbia Dept Comp Sci Vancouver BC V6T 1Z4 Canada Ecole Polytech GERAD Montreal PQ H3C 3A7 Canada Ecole Polytech Dept Math & Ind Engn Montreal PQ H3C 3A7 Canada Stanford Univ Dept Management Sci & Engn Syst Optimizat Lab Stanford CA 94305 USA 

出 版 物：《SIAM JOURNAL ON SCIENTIFIC COMPUTING》 (工业与应用数学会科学计算杂志)

年 卷 期：2020年第42卷第3期

页      面：A1809-A1835页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：Office of Naval Research [N00014-17-1-2009] NSERC [299010-04] National Institute of General Medical Sciences of the National Institutes of Health [U01GM102098] 

主　　题：nonlinear programming exact penalty smooth penalty factorization-free iterative methods trust-region 

摘      要：We develop a general equality-constrained nonlinear optimization algorithm based on a smooth penalty function proposed by Fletcher in 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 at 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.