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Stability of Two-Stage Stochastic Programs Under Problem-Dependent Costs

Nils Peyrousset, Benoît Tran

TL;DR

A direct stability approach using the primal optimal transport formulation, which proves that under minimal regularity conditions and a regret domination property, the optimal value function remains Lipschitz continuous with respect to problem-dependent transport costs.

Abstract

Classical stability theory for stochastic programming relies on the Wasserstein-Fortet-Mourier duality, which requires the ground cost to be a distance. When using problem-dependent costs instead of metrics, this duality no longer yields Fortet-Mourier bounds. This paper develops a direct stability approach using the primal optimal transport formulation. We prove that under minimal regularity conditions and a regret domination property, the optimal value function remains Lipschitz continuous with respect to problem-dependent transport costs. Our approach works directly with transport couplings rather than relying on dual representations to establish stability bounds. We present two applications: (1) For linear programs with continuous second-stage, we show that regret domination holds with constants depending on dual bounds and Lipschitz properties, using sensitivity analysis. (2) For mixed-integer second-stage problems, we show that combinatorial structure can be exploited to obtain tight regret bounds. We analyze several examples as illustrations. These results provide theoretical justification for problem-dependent scenario reduction approaches and enable their application to both continuous and discrete stochastic programs.

Stability of Two-Stage Stochastic Programs Under Problem-Dependent Costs

TL;DR

A direct stability approach using the primal optimal transport formulation, which proves that under minimal regularity conditions and a regret domination property, the optimal value function remains Lipschitz continuous with respect to problem-dependent transport costs.

Abstract

Classical stability theory for stochastic programming relies on the Wasserstein-Fortet-Mourier duality, which requires the ground cost to be a distance. When using problem-dependent costs instead of metrics, this duality no longer yields Fortet-Mourier bounds. This paper develops a direct stability approach using the primal optimal transport formulation. We prove that under minimal regularity conditions and a regret domination property, the optimal value function remains Lipschitz continuous with respect to problem-dependent transport costs. Our approach works directly with transport couplings rather than relying on dual representations to establish stability bounds. We present two applications: (1) For linear programs with continuous second-stage, we show that regret domination holds with constants depending on dual bounds and Lipschitz properties, using sensitivity analysis. (2) For mixed-integer second-stage problems, we show that combinatorial structure can be exploited to obtain tight regret bounds. We analyze several examples as illustrations. These results provide theoretical justification for problem-dependent scenario reduction approaches and enable their application to both continuous and discrete stochastic programs.
Paper Structure (25 sections, 9 theorems, 79 equations)

This paper contains 25 sections, 9 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: Notation and Optimal Transport Duality
  3. Wasserstein Distances and Probability Metrics
  4. Fortet-Mourier Metrics
  5. Function Spaces and Regularity
  6. Stochastic Programming Notation
  7. Optimal Transport Duality and Fortet-Mourier Metrics
  8. General Kantorovich Duality
  9. The Distance Case: Wasserstein-Fortet-Mourier Correspondence
  10. The Problem-Dependent Case: Breaking the Duality
  11. Problem-Dependent Ground Costs
  12. General Framework
  13. Examples of Problem-Dependent Costs
  14. Main Stability Result: Regret Domination implies Stability
  15. Regret and Regret Domination
  16. ...and 10 more sections

Key Result

Theorem 2.1

For probability measures $\mu, \nu$ on a separable metric space $(\Xi, d)$, where $W_1$ is the 1-Wasserstein distance and $\zeta_1$ is the first-order Fortet-Mourier metric.

Theorems & Definitions (29)

  • Theorem 2.1: 1-Wasserstein equals 1-Fortet-Mourier
  • proof
  • Lemma 2.2: Wasserstein Distance Ordering
  • proof
  • Corollary 2.3: Classical Stability Bounds via Wasserstein Distances
  • proof
  • Definition 3.1: Problem-Dependent Ground Cost
  • Example 3.2: Bertsimas-Mundru Cost
  • Example 3.3: Inventory Management
  • Example 3.4: Average Regret Cost
  • ...and 19 more