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On the Sparsity of Optimal Linear Decision Rules for a Class of Robust Optimization Problems with Box Uncertainty Sets

Haihao Lu, Brad Sturt

TL;DR

This paper proves that for a broad class of robust production–inventory problems with box uncertainty, there exists an optimal linear decision rule whose nonzero parameters grow only linearly with the planning horizon, despite the nominal quadratic parameter count. It introduces a sparsity-exploiting reformulation and an active-set algorithm to identify the nonzero parameters, achieving substantial computational speedups on problems with hundreds of periods. The authors show the approach scales to large instances and extend sparsity results to dynamic newsvendor problems and non-box uncertainty sets, with detailed proofs grounded in extreme-point analysis. The practical impact is significant: enabling near-optimal robust policies at scales previously intractable and guiding algorithm design that leverages sparsity in dynamic robust optimization.

Abstract

We consider a class of production-inventory problems with box uncertainty sets from the seminal work of Ben-Tal et al. (2004) on linear decision rules in robust optimization. We prove that there always exists an optimal linear decision rule for this class of problems in which the number of nonzero parameters in the linear decision rule grows linearly in the number of time periods. This is the first result to prove that optimal linear decision rules are sparse in a widely-studied class of robust optimization problems with many time periods. Harnessing this sparsity guarantee, we introduce a reformulation technique that allows robust optimization problems such as production-inventory problems to be solved as a compact linear optimization problem when most of the parameters of the linear decision rules are forced to be equal to zero. We also develop an active set method for identifying the parameters of linear decision rules that are equal to zero at optimality. In numerical experiments on production-inventory problems with hundreds of time periods, we find that our reformulation technique coupled with the active set method yield more than a 32x speedup over state-of-the-art linear optimization solvers in computing linear decision rules that are within 1\% of optimal. Our proofs and algorithms are based on a principled analysis of extreme points of linear optimization formulations.

On the Sparsity of Optimal Linear Decision Rules for a Class of Robust Optimization Problems with Box Uncertainty Sets

TL;DR

This paper proves that for a broad class of robust production–inventory problems with box uncertainty, there exists an optimal linear decision rule whose nonzero parameters grow only linearly with the planning horizon, despite the nominal quadratic parameter count. It introduces a sparsity-exploiting reformulation and an active-set algorithm to identify the nonzero parameters, achieving substantial computational speedups on problems with hundreds of periods. The authors show the approach scales to large instances and extend sparsity results to dynamic newsvendor problems and non-box uncertainty sets, with detailed proofs grounded in extreme-point analysis. The practical impact is significant: enabling near-optimal robust policies at scales previously intractable and guiding algorithm design that leverages sparsity in dynamic robust optimization.

Abstract

We consider a class of production-inventory problems with box uncertainty sets from the seminal work of Ben-Tal et al. (2004) on linear decision rules in robust optimization. We prove that there always exists an optimal linear decision rule for this class of problems in which the number of nonzero parameters in the linear decision rule grows linearly in the number of time periods. This is the first result to prove that optimal linear decision rules are sparse in a widely-studied class of robust optimization problems with many time periods. Harnessing this sparsity guarantee, we introduce a reformulation technique that allows robust optimization problems such as production-inventory problems to be solved as a compact linear optimization problem when most of the parameters of the linear decision rules are forced to be equal to zero. We also develop an active set method for identifying the parameters of linear decision rules that are equal to zero at optimality. In numerical experiments on production-inventory problems with hundreds of time periods, we find that our reformulation technique coupled with the active set method yield more than a 32x speedup over state-of-the-art linear optimization solvers in computing linear decision rules that are within 1\% of optimal. Our proofs and algorithms are based on a principled analysis of extreme points of linear optimization formulations.
Paper Structure (27 sections, 16 theorems, 111 equations, 6 figures, 1 algorithm)

This paper contains 27 sections, 16 theorems, 111 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and Problem Setting
  3. Robust Optimization and Linear Decision Rules
  4. The Production-Inventory Problem
  5. Main Theoretical Result
  6. The Value of Sparsity in Algorithm Design
  7. Preliminaries
  8. Compact Reformulation For Fixed Active Set
  9. Active Set Method
  10. Numerical Experiments
  11. Extensions and Limitations
  12. Dynamic Newsvendor Problem.
  13. Inventory Management with Non-Box Uncertainty Sets.
  14. Conclusion and Future Research
  15. Proofs of Lemmas \ref{['lem:1']}, \ref{['lem:2']}, and \ref{['lem:zeros']}
  16. ...and 12 more sections

Key Result

Theorem 1

Consider a cost function of the form prob:example1:a-prob:example1:d and let Assumption ass:1 hold. Suppose that $c_{te}>0$ for every $t\in [T]$ and $e\in [E]$, and let $\delta \triangleq\min_{e\in [E]} \delta_e$ denote the minimum lead time. Then there exists an optimal solution $\bar{{\bf y}}$ for

Figures (6)

  • Figure 1: Computation times for active set method and robust counterpart with $E = 5$ factories.
  • Figure 2: Objective value and computation time for active set method with $T = 240$ time periods.
  • Figure EC.1: Sparsity of optimal linear decision rules for production-inventory problem, $E = 3$.
  • Figure EC.2: Sparsity of optimal linear decision rules for production-inventory problem, $E = 4$.
  • Figure EC.3: Sparsity of optimal linear decision rules for production-inventory problem, $E = 5$.
  • ...and 1 more figures

Theorems & Definitions (37)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Proposition 1
  • Lemma 4
  • Proposition 2
  • Remark 1
  • Remark 2
  • Remark 3
  • ...and 27 more