Approximations and solution estimates in 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/s10107-017-1165-0
• 发表日期: 2018
• 页码: 479-506
• 关键词: lower semicontinuous functions; epi-distance topology; convex-sets cones; error-bounds; quantitative stability; variational systems; optimality conditions; nonsmooth analysis; tilt stability; CONVERGENCE

摘要

Approximation is central to many optimization problems and the supporting theory provides insight as well as foundation for algorithms. In this paper, we lay out a broad framework for quantifying approximations by viewing finite- and infinite-dimensional constrained minimization problems as instances of extended real-valued lower semicontinuous functions defined on a general metric space. Since the Attouch-Wets distance between such functions quantifies epi-convergence, we are able to obtain estimates of optimal solutions and optimal values through bounds of that distance. In particular, we show that near-optimal and near-feasible solutions are effectively Lipschitz continuous with modulus one in this distance. Under additional assumptions on the underlying metric space, we construct approximating functions involving only a finite number of parameters that still are close to an arbitrary extended real-valued lower semicontinuous functions.