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Finite adaptability in two-stage robust optimization: asymptotic optimality and tractability

Safia Kedad-Sidhoum, Anton Medvedev, Frédéric Meunier

TL;DR

The paper studies finite adaptability in two-stage robust optimization and its asymptotic behavior as k grows, aiming to bridge complete adaptability and computational tractability. It demonstrates that the classical continuity assumption may fail and provides counterexamples. It introduces a stronger modified continuity condition and proves convergence lim_{k->infty} val(fin-adapt) = val(comp-adapt) for compact Ω and affine dependence. Under deterministic A and B and polyhedral uncertainty with bounded vertices, it establishes polynomial-time solvability for k ≤ 3 via geometric coverings and LP/MILP reformulations, including an explicit MILP for k=2, and contributes new polytope covering results with potential broader impact.

Abstract

Two-stage robust optimization is a fundamental paradigm for modeling and solving optimization problems with uncertain parameters. A now classical method within this paradigm is finite adaptability, introduced by Bertsimas and Caramanis (IEEE Transactions on Automatic Control, 2010). It consists in restricting the recourse to a finite number $k$ of possible values. In this work, we point out that the continuity assumption they stated to ensure the convergence of the method when $k$ goes to infinity is not correct, and we propose an alternative assumption for which we prove the desired convergence. Bertsimas and Caramanis also established that finite adaptability is NP-hard, even in the special case when $k=2$, the variables are continuous, and only specific parameters are subject to uncertainty. We provide a theorem showing that this special case becomes polynomial when the uncertainty set is a polytope with a bounded number of vertices, and we extend this theorem for $k=3$ as well. On our way, we establish new geometric results on coverings of polytopes with convex sets, which might be interesting for their own sake.

Finite adaptability in two-stage robust optimization: asymptotic optimality and tractability

TL;DR

The paper studies finite adaptability in two-stage robust optimization and its asymptotic behavior as k grows, aiming to bridge complete adaptability and computational tractability. It demonstrates that the classical continuity assumption may fail and provides counterexamples. It introduces a stronger modified continuity condition and proves convergence lim_{k->infty} val(fin-adapt) = val(comp-adapt) for compact Ω and affine dependence. Under deterministic A and B and polyhedral uncertainty with bounded vertices, it establishes polynomial-time solvability for k ≤ 3 via geometric coverings and LP/MILP reformulations, including an explicit MILP for k=2, and contributes new polytope covering results with potential broader impact.

Abstract

Two-stage robust optimization is a fundamental paradigm for modeling and solving optimization problems with uncertain parameters. A now classical method within this paradigm is finite adaptability, introduced by Bertsimas and Caramanis (IEEE Transactions on Automatic Control, 2010). It consists in restricting the recourse to a finite number of possible values. In this work, we point out that the continuity assumption they stated to ensure the convergence of the method when goes to infinity is not correct, and we propose an alternative assumption for which we prove the desired convergence. Bertsimas and Caramanis also established that finite adaptability is NP-hard, even in the special case when , the variables are continuous, and only specific parameters are subject to uncertainty. We provide a theorem showing that this special case becomes polynomial when the uncertainty set is a polytope with a bounded number of vertices, and we extend this theorem for as well. On our way, we establish new geometric results on coverings of polytopes with convex sets, which might be interesting for their own sake.
Paper Structure (11 sections, 12 theorems, 23 equations, 1 figure)

This paper contains 11 sections, 12 theorems, 23 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Asymptotic optimality
  3. Tractable cases
  4. Geometric preliminaries
  5. Cover, partition, and convexity
  6. Covering polytopes with closed sets
  7. Covering polytopes with three convex sets
  8. Algorithms
  9. When $\Omega$ is one-dimensional
  10. When $k=2$ and $k=3$
  11. An explicit mixed integer linear program for $k=2$

Key Result

Theorem 1.1

Suppose that $k=2$. Then prob:finite-adapt with $A(\cdot)$ and $B(\cdot)$ deterministic can be solved by at most $3^{|V(\Omega)|}$ resolutions of a linear program of polynomial size (or a mixed linear program in case of integral variables).

Figures (1)

  • Figure 1: Representation of an optimal cover of problem \ref{['prob:cover_no_partition']}

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Example 1: Counter-example with a finite gap
  • Example 2: Counter-example with an infinite gap
  • Theorem 2.1
  • proof
  • Remark 1
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['lem:convexity']}
  • ...and 15 more