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Convergence and Bound Computation for Chance Constrained Distributionally Robust Models using Sample Approximation

Jiaqi Lei, Sanjay Mehrotra

TL;DR

The paper addresses convergence and bound computation for distributionally robust chance-constrained models under general ambiguity sets, demonstrating that a sample-based discretization of the ambiguity set yields a convergent approximation to the original DRO model. It establishes explicit bounds on constraint satisfaction differences and the optimal value gap, and provides practical methods to compute confidence intervals for the approximated objective. The work also verifies that concrete ambiguity-set families—moment-based, mean-variance, and $l_n$-Wasserstein—satisfy the key discretization assumption, and it delivers rate results and operational guidance for implementing the approach. This yields a tractable, data-driven pathway to robust decision-making with quantifiable out-of-sample performance in settings where both the objective and constraints are uncertain.

Abstract

This paper considers a distributionally robust chance constraint model with a general ambiguity set. We show that a sample based approximation of this model converges under suitable sufficient conditions. We also show that upper and lower bounds on the optimal value of the model can be estimated statistically. Specific ambiguity sets are discussed as examples.

Convergence and Bound Computation for Chance Constrained Distributionally Robust Models using Sample Approximation

TL;DR

The paper addresses convergence and bound computation for distributionally robust chance-constrained models under general ambiguity sets, demonstrating that a sample-based discretization of the ambiguity set yields a convergent approximation to the original DRO model. It establishes explicit bounds on constraint satisfaction differences and the optimal value gap, and provides practical methods to compute confidence intervals for the approximated objective. The work also verifies that concrete ambiguity-set families—moment-based, mean-variance, and -Wasserstein—satisfy the key discretization assumption, and it delivers rate results and operational guidance for implementing the approach. This yields a tractable, data-driven pathway to robust decision-making with quantifiable out-of-sample performance in settings where both the objective and constraints are uncertain.

Abstract

This paper considers a distributionally robust chance constraint model with a general ambiguity set. We show that a sample based approximation of this model converges under suitable sufficient conditions. We also show that upper and lower bounds on the optimal value of the model can be estimated statistically. Specific ambiguity sets are discussed as examples.
Paper Structure (13 sections, 13 theorems, 60 equations)

This paper contains 13 sections, 13 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Contributions
  4. Model Approximation and Known Technical Results
  5. Additional Definitions and Assumptions
  6. Technical Assumption
  7. Known Results
  8. New Convergence Results
  9. Upper and Lower Bounds
  10. Examples
  11. Moment-Based Ambiguity Set
  12. Mean and Variance Ambiguity Set
  13. $l_n$-Wasserstein Ambiguity Set

Key Result

Proposition 1

Liu19zhigljavsky2007stochastic Let $\xi_1, \xi_2,\dots,\xi_\omega$ be uniformly distributed on the support $\Xi\subseteq\mathbb{R}^d$. Then $\beta_{|\Omega|}$ follows an extreme value distribution with where $w.p.1$ means with probability 1.

Theorems & Definitions (21)

  • definition 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • ...and 11 more