Distributionally robust optimization with polynomial densities: theory, models and algorithms

TL;DR

This paper introduces a new class of ambiguity sets for distributionally robust optimization using polynomial densities, enabling more realistic modeling and computational tractability for complex distributional information.

Contribution

It proposes ambiguity sets with sum-of-squares polynomial densities, enhancing expressiveness and tractability in distributionally robust optimization models.

Findings

Ambiguity sets with polynomial densities can incorporate higher-order moments and conditional distributions.

Certain worst-case expectation constraints are computationally tractable with these new ambiguity sets.

Practical applications demonstrated in portfolio optimization and insurance risk aggregation.

Abstract

In distributionally robust optimization the probability distribution of the uncertain problem parameters is itself uncertain, and a fictitious adversary, e.g., nature, chooses the worst distribution from within a known ambiguity set. A common shortcoming of most existing distributionally robust optimization models is that their ambiguity sets contain pathological discrete distribution that give nature too much freedom to inflict damage. We thus introduce a new class of ambiguity sets that contain only distributions with sum-of-squares polynomial density functions of known degrees. We show that these ambiguity sets are highly expressive as they conveniently accommodate distributional information about higher-order moments, conditional probabilities, conditional moments or marginal distributions. Exploiting the theoretical properties of a measure-based hierarchy for polynomial optimization due to Lasserre [SIAM J. Optim. 21(3) (2011), pp. 864--885], we prove that certain worst-case expectation constraints are computationally tractable under these new ambiguity sets. We showcase the practical applicability of the proposed approach in the context of a stylized portfolio optimization problem and a risk aggregation problem of an insurance company.