Scenario-based Regularization: A Tractable Framework for Distributionally Robust Stochastic Optimization

TL;DR

This work introduces Scenario-Based Regularized SAA (SBR-SAA), a tractable, gradient-norm regularization framework for stochastic optimization that leverages a selected set of scenarios to explicitly control sensitivity to perturbations in $oldsymbol{\xi}$. By establishing an exact equivalence to a decision-dependent WDRO with a perturbed center and radius, the authors deliver WDRO-like finite-sample guarantees and asymptotic consistency while avoiding full min-max reformulations. They demonstrate the framework in two applications: a multi-product newsvendor where SBR-SAA provides a computationally efficient surrogate with competitive out-of-sample performance, and a mean-risk portfolio problem where incorporating adverse scenarios enhances tail performance via both quadratic- and linear-aggregation variants. Theoretical results are complemented by comprehensive numerical experiments, including MISOCP reformulations that enable scalable optimization, and case studies that illustrate when targeted scenario robustness can outperform or complement standard WDRO. Overall, SBR-SAA offers a transparent, flexible, and practically effective approach to distributional robustness in data-driven stochastic optimization.

Abstract

We propose a flexible scenario-based regularized Sample Average Approximation (SBR-SAA) framework for stochastic optimization. This work is motivated by challenges in standard Wasserstein Distributionally Robust Optimization (WDRO), where out-of-sample performance, particularly tail risk, is sensitive to the choice of the p-norm, and formulations can be computationally intractable. Our method is inspired by the asymptotic expansion of the WDRO objective and introduces a regularizer that penalizes the (sub)gradient norm of the objective at a selected set of scenarios. This framework serves a dual purpose: (i) it provides a computationally tractable alternative to WDRO by using a representative subset of the data, and (ii) it can provide targeted robustness by incorporating user-defined adverse scenarios. We establish the theoretical properties of this framework by proving its equivalence to a decision-dependent WDRO problem, from which we derive finite sample guarantees and asymptotic consistency. We demonstrate the method's efficacy in two applications: (1) a multi-product newsvendor problem, where SBR-SAA serves as a tractable alternative to NP-hard WDRO, and (2) a mean-risk portfolio optimization problem, where it successfully uses historical crisis data to improve out-of-sample performance.

Figures (10)

• Figure 1.1: Out-of-sample performance of $F(\hat{x}_{1},\xi)$ (1-WDRO) and $F(\hat{x}_{2},\xi)$ (2-WDRO) for a fixed training sample. Panel (a) shows individual out-of-sample evaluations; panel (b) compares their boxplots. Both methods display similar average performance, but 2-WDRO exhibits noticeably larger extreme values.
• Figure 1.2: Comparison of 1-WDRO and 2-WDRO over multiple training samples and different radii $\varepsilon$. Panel (a) shows tubes between the 10% and 90% quantiles and the corresponding means of the out-of-sample expected value. Panel (b) reports the tubes and means for the out-of-sample difference between CVaR$_{\alpha}$ (with $\alpha=5\%$) and the mean. While both methods exhibit similar expected performance, 2-WDRO systematically attains larger CVaR$_\alpha$-to-mean gaps, indicating higher tail risk.
• Figure 4.1: Out-of-sample performance curves $\hat{\mathcal{C}}_{m,n}$ for $n=50$ and $m=5$ (5-SBR-SAA) and $m=15$ (15-SBR-SAA), based on a single training sample. The y-axis represents the out-of-sample expected cost, while the x-axis represents the out-of-sample tail risk premium (CVaR$_{5\%}$ - Mean).
• Figure 4.2: Out-of-sample performance of SBR-SAA relative to SAA across 200 replications ($n=50$). Each point represents the relative performance of the SBR-SAA solution that minimized the out-of-sample expected cost for a given sample. Negative values on both axes indicate an improvement over SAA. (a) $m=5$ scenarios. (b) $m=15$ scenarios.
• Figure 4.3: Daily returns of the S&P 500 index in 2020, with thresholds at $-2\%$, $-3.5\%$, and $-5\%$ used to identify adverse market days for Case Study 1.

Theorems & Definitions (33)

• Example 1.1: Motivating example: the role of the Wasserstein order
• Definition 2.1: Wasserstein distance
• Theorem 2.1: Gao2016Blanchet2019
• Lemma 2.1: Taylor expansion of worst-case expectation
• Theorem 3.1
• proof
• Lemma 3.1
• Corollary 3.1
• Theorem 3.2: Kantorovich–Rubinstein Duality
• Lemma 3.2