Robust Constructions of Risk Measures for Optimization under Uncertainty

For ambiguous probability distributions, when the probability measure is described by an uncertainty set, robust constructions of risk measures are proposed. The apparatus of polyhedral coherent risk measures is considered. The calculation of robust constructions of such risk measures for an uncertainty polyhedral set are reduced to appropriate linear programming problems. Portfolio optimization problems on the reward-risk ratio, in which the reward and risk are described by robust constructions of the average return and a polyhedral coherent risk measure respectively, are considered as well. It is shown how such problems are reduced to appropriate linear programming problems. Portfolio optimization problems for imprecise scenario estimates of random variables and use of the apparatus of polyhedral coherent risk measure for them are discussed as well.