Robust Data-Driven CARA Optimization

Abstract

We focus on data-driven, risk-averse optimization problems where decision-makers exhibit constant absolute risk aversion (CARA), represented by an exponential utility function. We consider payoff functions expressible as conic optimization problems, highlighting their expansive use in prescriptive analytics. Aiming to mitigate the overfitting issues inherent in empirical distribution-based optimization, we employ a robust satisficing strategy, which combines a target parameter with a Wasserstein distance metric, to define acceptable solutions, robustly. This target parameter could be determined via cross validation to mitigate overfitting and improve out-of-sample performance. Nevertheless, integrating an exponential utility function into a robust satisficing framework introduces substantial computational challenges. Traditional convex approximations, despite being safe and tractable, do not guarantee feasibility when applied to some appropriate target levels. We overcome this inconsistency by delineating specific conditions, namely, a scenario with complete and bounded recourse, polyhedral uncertainty support and polyhedral metric norm, under which a safe, tractable, and consistent approximation of the data-driven robust satisficing problem can be guaranteed for any reasonably specified target. To ensure compliance with the complete and bounded recourse conditions, we subsequently introduce the notion of augmented exponential utility by restricting the domain of the utility function. We validate our theoretical results with computational experiments in two applications: data-driven portfolio optimization and an optimal location problem. The numerical results underscore the efficacy of incorporating robustness into optimization models, significantly enhancing solution quality as compared to traditional empirical optimization approaches.

Keywords: robust optimization, distributionally robust optimization, conic optimization, exponential utility, robust satisficing