Network-Based Approximate Linear Programming for Discrete Optimization

55 Pages Posted: 7 Dec 2017 Last revised: 12 Jun 2019

See all articles by Selvaprabu Nadarajah

Selvaprabu Nadarajah

University of Illinois at Chicago - College of Business Administration

Andre Augusto Cire

University of Toronto at Scarborough - Division of Management; University of Toronto - Operations Management

Date Written: December 3, 2017

Abstract

We present relaxations for discrete optimization problems using approximate linear programs (ALPs) defined on multiple networks that represent different state-space aggregations. Our network ALP leverages information across these networks using a piecewise-constant value function approximation and its optimistic bound is theoretically guaranteed to weakly improve upon the bounds from the individual networks used in its construction. Solving network ALPs is challenging due to its large number of constraints -- a well-known issue when applying approximate linear programming. We side-step this issue by using a clique-based graph representation to design a network ALP restriction that admits a polynomial-time solvable extended formulation, which we show to also deliver a weakly better bound than individual networks. We execute a computational study on a challenging bilinear problem arising in marketing analytics and a routing application encountered in the preemptive maintenance of energy or city-owned assets. When used within a branch-and-bound scheme, the network ALP restriction significantly outperforms in both solution quality and time the following benchmarks: a state-of-the-art mathematical programming solver, a generic aggregation/dissaggregation scheme applied to a single network, and a heuristic that chooses the best bound among individual networks.

Keywords: Dynamic Programming, Approximate Linear Programming, Discrete Optimization

Suggested Citation

Nadarajah, Selvaprabu and Cire, Andre Augusto, Network-Based Approximate Linear Programming for Discrete Optimization (December 3, 2017). Available at SSRN: https://ssrn.com/abstract=3081898 or http://dx.doi.org/10.2139/ssrn.3081898

Selvaprabu Nadarajah (Contact Author)

University of Illinois at Chicago - College of Business Administration ( email )

601 South Morgan Street
Chicago, IL 60607
United States

Andre Augusto Cire

University of Toronto at Scarborough - Division of Management ( email )

1265 Military Trial
Scarborough, Ontario M1C 1A4
Canada

University of Toronto - Operations Management ( email )

105 St. George st
Toronto, ON M5S 3E6
Canada

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