A statistical perspective on linear programs with uncertain parameters

Abstract

We consider linear programs where some parameters in the objective functions are unknown but data are available. For a risk-averse modeler, the solutions of these linear programs should be picked in a way that can perform well for a range of likely scenarios inferred from the data. The conventional approach uses robust optimization. Taking the optimality gap as our loss criterion, we argue that this approach can be high-risk, in the sense that the optimality gap can be large with significant probability. We then propose two computationally tractable alternatives: The first uses bootstrap aggregation, or so-called bagging in the statistical learning literature, while the second uses Bayes estimator in the decision-theoretic framework. Both are simulation-based schemes that aim to improve the distributional behavior of the optimality gap by reducing its frequency of hitting large values.