Contextual Quantile Minimization for Two-Stage Stochastic Programs

TL;DR

This work tackles risk-averse decision-making under contextual information in two-stage stochastic programs by formulating a contextual quantile minimization problem using a generic estimator for the conditional quantile $Q_\tau(f(x,\xi))$ given context $Z$. It establishes theoretical guarantees, proving almost-sure convergence and convergence in probability of the optimal value and solutions under mild regularity, and introduces the stochastic inexact constraint generation (SiCG) algorithm to efficiently solve nonconvex quantile-based two-stage problems. The authors demonstrate computational and managerial advantages through a single-server appointment scheduling experiment, showing that incorporating contextual information and a quantile objective yields risk-averse schedules with improved tail performance and robustness to distributional mis-specification. The framework combines flexible conditional quantile estimation with a scalable decomposition method, offering a data-driven, risk-aware approach applicable to general two-stage stochastic programs and settings where context is available ahead of decisions.

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.

Figures (23)

• Figure 1: Example of the estimated quantile function and its domain partition
• Figure 2: Percentage of instances (over $30$ replications) solved within $1$ hour time limit using SiCG and MILP for $\tau\in\{0.95,0.90\}$ when $n=6$ and $N=1000$.
• Figure 3: Optimal appointment schedules from three different models when $\nu=0.2$ and $R=0.5$ under a quantile objective.
• Figure 4: Optimal appointment schedules from three different models when $\nu=0.2$ and $R=0.5$ under an expectation objective.
• Figure 5: Out-of-sample 95% percentiles of the total cost over $20$ optimal schedules under a quantile objective from three different models when $\nu=0.2$ and $R=0.5$.

Theorems & Definitions (16)

• Example 3.1: Local Regression Estimate
• Example 4.1
• Theorem 1
• Theorem 2
• Example 4.2: Kernel Regression
• Example 4.3: $k$NN Regression
• Lemma 1
• Theorem 3
• Theorem 4
• Remark 1