Assessing solution quality in risk-averse stochastic programs

TL;DR

This paper develops new unbiased estimators for assessing solution quality in risk-averse stochastic programs, addressing bias issues in traditional methods and extending estimation techniques to risk-averse contexts.

Contribution

It introduces a novel approach using two independent samples to produce unbiased optimality gap estimates for risk-averse problems, expanding existing methods.

Findings

The proposed estimators are unbiased for spectral and quadrangle risk measures.

The method is tractable and improves the quality of optimality gap estimates.

It can incorporate bias and variance reduction techniques.

Abstract

In optimization problems, the quality of a candidate solution can be characterized by the optimality gap. For most stochastic optimization problems, this gap must be statistically estimated. We show that for risk-averse problems, standard estimators are optimistically biased, which compromises the statistical guarantee on the optimality gap. We introduce estimators for risk-averse problems that do not suffer from this bias. Our method relies on using two independent samples, each estimating a different component of the optimality gap. Our approach extends a broad class of optimality gap estimation methods from the risk-neutral case to the risk-averse case, such as the multiple replications procedure and its one- and two-sample variants. We show that our approach is tractable and leads to high-quality optimality gap estimates for spectral and quadrangle risk measures. Our approach can further make use of existing bias and variance reduction techniques.