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A fast approximate column-and-constraint generation method for two-stage robust mixed-integer programs

Marc Goerigk, Dorothee Henke, Johannes Kager, Fabian Schäfer, Clemens Thielen

TL;DR

Addresses two-stage robust mixed-integer programs with finite scenario sets and introduces a fast approximate column-and-constraint generation method (ASBP). ASBP integrates dual bounds, adaptive time limits, and gap propagation to quickly bracket worst-case scenarios and can solve the master problem up to a non-zero target gap. The authors prove correctness and termination and show that ASBP outperforms ISAM and SRP on robust capacitated location routing and BACASP, particularly when the second stage is hard. The results indicate a practical boost in solving large-scale two-stage robust optimization and point to opportunities for extension to nonlinear models and data-driven speedups.

Abstract

This paper presents a new column-and-constraint generation method for two-stage robust mixed-integer programs with finite uncertainty sets. Our method combines and extends speed-up techniques used in previous column-and-constraint generation methods and introduces several new techniques. In particular, it uses dual bounds for second-stage problems in order to allow a faster identification of the next promising scenario to be added to the master problem. Moreover, adaptive time limits are imposed to avoid getting stuck on particularly hard second-stage problems, and a gap propagation between master problem and second-stage problems is used to stop solving them earlier if only a given non-zero optimality gap is to be reached overall. This makes our method particularly effective for problems where solving the second-stage problem is computationally challenging. To evaluate the method's performance, we compare it to two recent column-and-constraint generation methods from the literature on two applications: a robust capacitated location routing problem and a robust integrated berth allocation and quay crane assignment and scheduling problem. The first problem features a particularly hard second stage, and we show that our method is able to solve considerably more and larger instances in a given time limit. Using the second problem, we verify the general applicability of our method, even for problems where the second stage is relatively easy.

A fast approximate column-and-constraint generation method for two-stage robust mixed-integer programs

TL;DR

Addresses two-stage robust mixed-integer programs with finite scenario sets and introduces a fast approximate column-and-constraint generation method (ASBP). ASBP integrates dual bounds, adaptive time limits, and gap propagation to quickly bracket worst-case scenarios and can solve the master problem up to a non-zero target gap. The authors prove correctness and termination and show that ASBP outperforms ISAM and SRP on robust capacitated location routing and BACASP, particularly when the second stage is hard. The results indicate a practical boost in solving large-scale two-stage robust optimization and point to opportunities for extension to nonlinear models and data-driven speedups.

Abstract

This paper presents a new column-and-constraint generation method for two-stage robust mixed-integer programs with finite uncertainty sets. Our method combines and extends speed-up techniques used in previous column-and-constraint generation methods and introduces several new techniques. In particular, it uses dual bounds for second-stage problems in order to allow a faster identification of the next promising scenario to be added to the master problem. Moreover, adaptive time limits are imposed to avoid getting stuck on particularly hard second-stage problems, and a gap propagation between master problem and second-stage problems is used to stop solving them earlier if only a given non-zero optimality gap is to be reached overall. This makes our method particularly effective for problems where solving the second-stage problem is computationally challenging. To evaluate the method's performance, we compare it to two recent column-and-constraint generation methods from the literature on two applications: a robust capacitated location routing problem and a robust integrated berth allocation and quay crane assignment and scheduling problem. The first problem features a particularly hard second stage, and we show that our method is able to solve considerably more and larger instances in a given time limit. Using the second problem, we verify the general applicability of our method, even for problems where the second stage is relatively easy.
Paper Structure (26 sections, 5 theorems, 18 equations, 4 figures, 4 tables, 4 algorithms)

This paper contains 26 sections, 5 theorems, 18 equations, 4 figures, 4 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related literature and our contribution
  3. Our contribution
  4. General structure and known C&CG methods
  5. The approximate scenario bracketing procedure
  6. Gap propagation in column-and-constraint generation methods
  7. Algorithm description
  8. Analysis of our algorithm
  9. Applications
  10. Robust capacitated location routing problem
  11. Problem description
  12. Instances
  13. Berth allocation and quay crane assignment and scheduling problem
  14. Problem description
  15. Instances
  16. ...and 11 more sections

Key Result

Proposition 2

Suppose that $\mathop{\mathrm{MAS}}\nolimits_D$ has been solved up to a gap $P \in [0, 1)$ for some subset $D\subseteq S$, and let $(\tilde{x}, \tilde{z})$ denote the obtained master solution. If $Q(\tilde{x}, s) \le \tilde{z}$ for all $s \in S$, then the first-stage solution $\tilde{x}$ is a soluti

Figures (4)

  • Figure 1: Performance plots of our ASBP with $\mu=0.5$ in blue, the ISAM toenissen11 in orange, and the SRP RODRIGUES2021499 in green for the RCLRP.
  • Figure 2: Performance plots of our ASBP with $\mu=0.5$ in blue, the ISAM toenissen11 in orange, and the SRP RODRIGUES2021499 in green for the BACASP.
  • Figure 3: Performance plots of our ASBP with different master-gap factors $\mu$.
  • Figure 4: Comparison of performance plots for different algorithms (ASBP in blue, ISAM in orange, SRP in green) shown once with the standard heuristic (solid lines) and once without heuristic (dashed lines). All other parameters are the same as for the small instances in Figures 1 and 2 of the main paper.

Theorems & Definitions (11)

  • Definition 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Lemma 5
  • proof
  • Theorem 6
  • proof
  • Proposition 7
  • ...and 1 more