Discrete Nonlinear Optimization by State-Space Decompositions
研究将带线性约束的二元非线性优化问题分解为多个低维动态规划模型，每个模型等价为最短路径问题，再通过混合整数线性规划关联得到精确解，并给出松弛机制以处理大状态空间，在收益管理、投资组合和医疗等应用中大幅优于现有方法。
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. Data and the online appendix are available at https://doi.org/10.1287/mnsc.2017.2849 . This paper was accepted by Yinyu Ye, optimization.