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

• 成果类型: Article
• 署名作者: de Klerk, Etienne; Kuhn, Daniel; Postek, Krzysztof
• 署名单位: Tilburg University; Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne; Erasmus University Rotterdam - Excl Erasmus MC; Erasmus University Rotterdam
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-019-01429-5
• 发表日期: 2020
• 页码: 265-296
• 关键词: semidefinite programming approach; bounds; inequalities; uncertainty

摘要

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 knownambiguity set. A common shortcoming of most existing distributionally robust optimization models is that their ambiguity sets contain pathological discrete distributions 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 (SOS) 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):864-885,2011), we prove that certain worst-case expectation constraints are polynomial-time solvable under these new ambiguity sets. We also show how SOS densities can be used to approximately solve thegeneral problem of moments. We showcase the applicability of the proposed approach in the context of a stylized portfolio optimization problem and a risk aggregation problem of an insurance company.