Discrete Nonlinear Optimization by State-Space Decompositions

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 repre...

Full description

Saved in:
Bibliographic Details
Published inManagement science Vol. 64; no. 10; pp. 4700 - 4720
Main Authors Bergman, David, Cire, Andre A.
Format Journal Article
LanguageEnglish
Published Linthicum INFORMS 01.10.2018
Institute for Operations Research and the Management Sciences
Subjects
algorithms
Analysis
Artificial intelligence
Dynamic programming
finite state
Health services administration
Heuristic
integer
Linear programming
Mathematical optimization
network-graphs
nonlinear
optimal control
Optimization
Portfolio management
programming
United States
Online AccessGet full text

Cover

More Information
Summary: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.
ISSN:0025-1909
1526-5501
DOI:10.1287/mnsc.2017.2849