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Network Flow Models for Robust Binary Optimization with Selective Adaptability

Merve Bodur, Timothy C. Y. Chan, Ian Yihang Zhu

TL;DR

This paper proposes new reformulation techniques for adaptive robust binary optimization (ARBO) problems with objective uncertainty with selective adaptability, and shows that these models can generate high-quality solutions and dual bounds in significantly less time than popular benchmark methods.

Abstract

Adaptive robust optimization problems have received significant attention in recent years, but remain notoriously difficult to solve when recourse decisions are discrete in nature. In this paper, we propose new reformulation techniques for adaptive robust binary optimization (ARBO) problems with objective uncertainty. Without loss of generality, we focus on ARBO problems with "selective adaptability", a term we coin to describe a common class of linking constraints between first-stage and second-stage solutions. Our main contribution revolves around a collection of exact and approximate network flow reformulations for the ARBO problem, which we develop by building upon ideas from the decision diagram literature. Our proposed models can generate feasible solutions, primal bounds and dual bounds, while their size and approximation quality can be precisely controlled through user-specified parameters. Furthermore, and in contrast with existing solution methods, these models are easy to implement and can be solved directly with standard off-the-shelf solvers. Through an extensive set of computational experiments, we show that our models can generate high-quality solutions and dual bounds in significantly less time than popular benchmark methods, often by orders of magnitude.

Network Flow Models for Robust Binary Optimization with Selective Adaptability

TL;DR

This paper proposes new reformulation techniques for adaptive robust binary optimization (ARBO) problems with objective uncertainty with selective adaptability, and shows that these models can generate high-quality solutions and dual bounds in significantly less time than popular benchmark methods.

Abstract

Adaptive robust optimization problems have received significant attention in recent years, but remain notoriously difficult to solve when recourse decisions are discrete in nature. In this paper, we propose new reformulation techniques for adaptive robust binary optimization (ARBO) problems with objective uncertainty. Without loss of generality, we focus on ARBO problems with "selective adaptability", a term we coin to describe a common class of linking constraints between first-stage and second-stage solutions. Our main contribution revolves around a collection of exact and approximate network flow reformulations for the ARBO problem, which we develop by building upon ideas from the decision diagram literature. Our proposed models can generate feasible solutions, primal bounds and dual bounds, while their size and approximation quality can be precisely controlled through user-specified parameters. Furthermore, and in contrast with existing solution methods, these models are easy to implement and can be solved directly with standard off-the-shelf solvers. Through an extensive set of computational experiments, we show that our models can generate high-quality solutions and dual bounds in significantly less time than popular benchmark methods, often by orders of magnitude.
Paper Structure (42 sections, 7 theorems, 45 equations, 4 figures, 7 tables, 2 algorithms)

This paper contains 42 sections, 7 theorems, 45 equations, 4 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Adaptive robust binary optimization
  4. Decision diagrams and network flow models
  5. Robust Binary Optimization with Selective Adaptability
  6. Problem definition
  7. Models with selective adaptability
  8. Preliminary model insights
  9. Constrained Network Flow Reformulations
  10. Reformulating the recourse feasible space
  11. Decision diagram formulation.
  12. Recourse network flow formulation.
  13. A complete network flow reformulation
  14. Network Flow Approximations
  15. Approximate decision diagrams
  16. ...and 27 more sections

Key Result

Lemma 1

$\mathop{\mathrm{Conv}}\nolimits(\mathcal{Y} \cap \mathcal{S}(\mathbf{x})) = \mathop{\mathrm{Conv}}\nolimits(\mathcal{Y}) \cap \mathcal{S}(\mathbf{x}), \ \ \forall \mathbf{x} \in \{0,1\}^m$.

Figures (4)

  • Figure 1: A decision diagram with six layers, where the zero-one label of each arc is indicated using a dashed or solid line.
  • Figure 2: The model-based optimality gap as a function of the total time to formulate and solve the network flow models. Note that the data points roughly follow an exponential decay curve, that is, an incremental increase in solution time can result in a large decrease in optimality gap, when solution times are small.
  • Figure 3: A robust assignment problem. Agents $1, \ldots, L$ are associated with weight $a_1, \ldots, a_L$, tasks $1, \ldots, M$ are associated with capacity $b_1, \ldots, b_M$, and agent-task pairings $(\ell,m)$ generate reward $\xi_{\ell m}$.
  • Figure 4: Model-based optimality gap of solutions of the Multi NF model.

Theorems & Definitions (14)

  • Definition 1: Selective Adaptability
  • Lemma 1
  • Proposition 1
  • Proposition 2
  • Remark 1
  • Remark 2
  • Example 1
  • Proposition 3
  • Example 2
  • Example 3
  • ...and 4 more