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

TL;DR

This paper develops a primal transport-based stability analysis for two-stage stochastic programs with problem-dependent costs, extending classical duality-based results to broader cost structures.

Contribution

It introduces a direct primal approach to stability, proving Lipschitz continuity of the value function under minimal conditions and applying it to both continuous and discrete stochastic programs.

Findings

Optimal value function is Lipschitz continuous with respect to problem-dependent costs.

Regret bounds depend on dual bounds and Lipschitz properties for linear programs.

Exploits combinatorial structure for tight bounds in mixed-integer problems.

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.