Contextual Quantile Minimization for Two-Stage Stochastic Programs

TL;DR

This paper introduces a novel risk-averse two-stage stochastic optimization framework that minimizes a quantile-based objective using contextual information, with proven convergence and a new solution method, demonstrated through scheduling experiments.

Contribution

It proposes a new quantile-based risk-averse optimization model with convergence analysis and a stochastic inexact constraint generation algorithm for two-stage problems.

Findings

Convergence of solutions under mild regularity conditions.

Effective computational performance demonstrated on scheduling problems.

Highlighting the importance of contextual information and risk attitudes.

Abstract

Contextual stochastic optimization is an advanced methodology to model uncertainty in the presence of contextual information during decision planning processes. Although classical methodologies focus on minimizing the expectation of a random loss, in many applications, risk-averse decision-makers may be interested in minimizing a specific quantile as a more prudent alternative. In this paper, we propose a new risk-averse contextual stochastic optimization problem with a quantile objective for general two-stage problems. Given historical data on the model's random parameters and contextual information, we model the conditional quantile by replacing the conditional expectation in its variational characterization with a generic estimator. Under two sets of mild regularity conditions, we derive the asymptotic almost-sure convergence and convergence in probability of the optimal solution and the optimal value of the associated optimization problem to their true counterparts. Optimization problems with a quantile objective is usually non-convex, which are generally regarded as challenging to solve. To address the computational difficulties, we propose a new stochastic inexact constraint generation method with convergence guarantee. Finally, through numerical experiments on a single-server appointment scheduling problem, we study the computational performance of our proposed solution method as well as operational performance of our proposed methodology. Our results demonstrate the importance of incorporating useful contextual information and decision-maker's risk attitude into the optimization model.