Consistent approximations in composite optimization

• 成果类型: Article
• 署名作者: Royset, Johannes O.
• 署名单位: United States Department of Defense; United States Navy; Naval Postgraduate School
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-022-01909-1
• 发表日期: 2023
• 页码: 339-372
• 关键词: quantitative stability; variational systems; tilt stability; nonsmooth; minimization; CONVERGENCE; optimality; convolution; algorithms; calmness

摘要

Approximations of optimization problems arise in computational procedures and sensitivity analysis. The resulting effect on solutions can be significant, with even small approximations of components of a problem translating into large errors in the solutions. We specify conditions under which approximations are well behaved in the sense of minimizers, stationary points, and level-sets and this leads to a framework of consistent approximations. The framework is developed for a broad class of composite problems, which are neither convex nor smooth. We demonstrate the framework using examples from stochastic optimization, neural-network based machine learning, distributionally robust optimization, penalty and augmented Lagrangian methods, interior-point methods, homotopy methods, smoothing methods, extended nonlinear programming, difference-of-convex programming, and multi-objective optimization. An enhanced proximal method illustrates the algorithmic possibilities. A quantitative analysis supplements the development by furnishing rates of convergence.