A proximal trust-region method for nonsmooth optimization with inexact function and gradient evaluations

• 成果类型: Article
• 署名作者: Baraldi, Robert J. J.; Kouri, Drew P. P.
• 署名单位: United States Department of Energy (DOE); Sandia National Laboratories
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-022-01915-3
• 发表日期: 2023
• 页码: 559-598
• 关键词: pde-constrained optimization; GLOBAL CONVERGENCE; nonconvex minimization; algorithms; SUM; complexity; shrinkage; sparsity

摘要

Many applications require minimizing the sum of smooth and nonsmooth functions. For example, basis pursuit denoising problems in data science require minimizing a measure of data misfit plus an l(1)-regularizer. Similar problems arise in the optimal control of partial differential equations (PDEs) when sparsity of the control is desired. We develop a novel trust-region method to minimize the sum of a smooth nonconvex function and a nonsmooth convex function. Our method is unique in that it permits and systematically controls the use of inexact objective function and derivative evaluations. When using a quadratic Taylor model for the trust-region subproblem, our algorithm is an inexact, matrix-free proximal Newton-type method that permits indefinite Hessians. We prove global convergence of our method in Hilbert space and demonstrate its efficacy on three examples from data science and PDE-constrained optimization.