Paper: arXiv 2408.15690

Authors: E. Ruben van Beesten, Nick W. Koning, David P. Morton

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.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical framework to solve the optimistic bias issue in optimality gap estimation for risk-averse stochastic programs using two independent samples. While the theoretical derivations, proofs, and methodological clarity are strong, the provided excerpt lacks empirical backtesting and real-world numerical experiments.

Research Flowchart

  flowchart TD
    A["Research Goal: Estimate optimality gap in risk-averse stochastic programs without optimistic bias"] --> B["Problem Identification: Standard risk-neutral estimators introduce optimistic bias in risk-averse settings"]
    B --> C["Inputs: Candidate solution, Risk measures (Spectral/Quadrangle), & Two independent sample sets"]
    C --> D["Core Method: Two-Sample Independent Estimator Structure"]
    D --> E["Computation: Apply MRP variants + Bias/Variance reduction techniques"]
    E --> F["Outcomes: Unbiased, tractable optimality gap estimates with valid statistical guarantees"]

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0 of 390 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.