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Learning Decision-Focused Uncertainty Sets in Robust Optimization

Irina Wang, Bart Van Parys, Bartolomeo Stellato

TL;DR

The paper tackles decision-making under uncertainty by learning contextual uncertainty sets that shape robust optimization problems. It formulates a constrained bi-level, decision-focused learning problem that minimizes a weighted performance metric while enforcing CVaR-based constraint satisfaction, solved via a stochastic augmented Lagrangian method with path-differentiable, nonsmooth analysis. The authors prove convergence to stationary points and establish finite-sample probabilistic guarantees, then demonstrate empirical gains over traditional robust and distributionally robust methods in portfolio optimization and inventory management. The work enables less conservative, context-aware robustness with theoretical guarantees and practical applicability, supported by an open-source codebase.

Abstract

We propose a data-driven technique to automatically learn contextual uncertainty sets in robust optimization, resulting in excellent worst-case and average-case performance while also guaranteeing constraint satisfaction. Our method reshapes the uncertainty sets by minimizing the expected performance across a contextual family of problems, subject to conditional-value-at-risk constraints. Our approach is very flexible, and can learn a wide variety of uncertainty sets while preserving tractability. We solve the constrained learning problem using a stochastic augmented Lagrangian method that relies on differentiating the solutions of the robust optimization problems with respect to the parameters of the uncertainty set. Due to the nonsmooth and nonconvex nature of the augmented Lagrangian function, we apply the nonsmooth conservative implicit function theorem to establish convergence to a critical point, which is a feasible solution of the constrained problem under mild assumptions. Using empirical process theory, we show finite-sample probabilistic guarantees of constraint satisfaction for the resulting solutions. Numerical experiments show that our method outperforms traditional approaches in robust and distributionally robust optimization in terms of out-of-sample performance and constraint satisfaction guarantees.

Learning Decision-Focused Uncertainty Sets in Robust Optimization

TL;DR

The paper tackles decision-making under uncertainty by learning contextual uncertainty sets that shape robust optimization problems. It formulates a constrained bi-level, decision-focused learning problem that minimizes a weighted performance metric while enforcing CVaR-based constraint satisfaction, solved via a stochastic augmented Lagrangian method with path-differentiable, nonsmooth analysis. The authors prove convergence to stationary points and establish finite-sample probabilistic guarantees, then demonstrate empirical gains over traditional robust and distributionally robust methods in portfolio optimization and inventory management. The work enables less conservative, context-aware robustness with theoretical guarantees and practical applicability, supported by an open-source codebase.

Abstract

We propose a data-driven technique to automatically learn contextual uncertainty sets in robust optimization, resulting in excellent worst-case and average-case performance while also guaranteeing constraint satisfaction. Our method reshapes the uncertainty sets by minimizing the expected performance across a contextual family of problems, subject to conditional-value-at-risk constraints. Our approach is very flexible, and can learn a wide variety of uncertainty sets while preserving tractability. We solve the constrained learning problem using a stochastic augmented Lagrangian method that relies on differentiating the solutions of the robust optimization problems with respect to the parameters of the uncertainty set. Due to the nonsmooth and nonconvex nature of the augmented Lagrangian function, we apply the nonsmooth conservative implicit function theorem to establish convergence to a critical point, which is a feasible solution of the constrained problem under mild assumptions. Using empirical process theory, we show finite-sample probabilistic guarantees of constraint satisfaction for the resulting solutions. Numerical experiments show that our method outperforms traditional approaches in robust and distributionally robust optimization in terms of out-of-sample performance and constraint satisfaction guarantees.
Paper Structure (59 sections, 5 theorems, 78 equations, 1 figure, 5 tables, 2 algorithms)

This paper contains 59 sections, 5 theorems, 78 equations, 1 figure, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Related work
  4. Data-driven robust optimization.
  5. Contextual optimization.
  6. Pareto efficiency in .
  7. Modeling languages for robust optimization.
  8. Differentiable convex optimization.
  9. Nonsmooth implicit differentiation.
  10. Risk measures.
  11. Layout of the paper
  12. The robust optimization problem
  13. Objective uncertainty and average performance
  14. Motivating example
  15. Problem setup.
  16. ...and 44 more sections

Key Result

Theorem 3.1

Suppose Assumption ass:unique holds. Let $z(\theta,x)$ be the unique solution, for a given input pair $(\theta, x)$, to the convex conic reformulation of the inner optimization problem eq:robust_prob. This solution is path differentiable with nonempty conservative Jacobian $J_{z}(\theta) \neq \empty

Figures (1)

  • Figure 1: Our high-level procedure. We evaluate the performance of the decision $z(\theta,x)$ in terms of the expected objective value and constraint satisfaction. The evaluation metric is defined in Section \ref{['sec:training_procedure']}.

Theorems & Definitions (7)

  • Definition 3.1: Conservative Jacobian path-diff
  • Theorem 3.1: Path differentiability of the solution
  • Proposition 3.1: Conservative Jacobians of the augmented Lagrangian \ref{['eq:lagrangian']}
  • Theorem 3.2
  • Theorem 4.1: Finite sample probabilistic guarantee
  • proof
  • Proposition C.1