Discrete Nonlinear Optimization by State-Space Decompositions

• 成果类型: Article
• 署名作者: Bergman, David; Cire, Andre A.
• 署名单位: University of Connecticut; University of Toronto; University Toronto Scarborough; University of Toronto
• 刊物名称: MANAGEMENT SCIENCE
• ISSN/ISSBN: 0025-1909
• DOI: 10.1287/mnsc.2017.2849
• 发表日期: 2018
• 页码: 4700-4720
• 关键词: nonlinear; algorithms; programming; integer; network-graphs; dynamic programming; optimal control; FINITE STATE

摘要

This paper investigates a decomposition approach for binary optimization problems with nonlinear objectives and linear constraints. Our methodology relies on the partition of the objective function into separate low-dimensional dynamic programming (DP) models, each of which can be equivalently represented as a shortest-path problem in an underlying state-transition graph. We show that the associated transition graphs can be related by a mixed-integer linear program (MILP) so as to produce exact solutions to the original nonlinear problem. To address DPs with large state spaces, we present a general relaxation mechanism that dynamically aggregates states during the construction of the transition graphs. The resulting MILP provides both lower and upper bounds to the nonlinear function, and it may be embedded in branch-and-bound procedures to find provably optimal solutions. We describe how to specialize our technique for structured objectives (e.g., submodular functions) and consider three problems arising in revenue management, portfolio optimization, and healthcare. Numerical studies indicate that the proposed technique often outperforms state-of-the-art approaches by orders of magnitude in these applications.