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

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