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Polyhedral coherent risk measures and their worst-case constructions with respect to the ambiguity set are considered. For the case of the discrete distribution and polyhedral ambiguity set, calculating such risk measures reduces to linear programming problems. The distributionally robust portfolio optimization problems based on the reward-risk ratio using worst-case constructions with respect to the polyhedral ambiguity set for these risk measures and average return are analyzed. They are reduced to the appropriate linear programming problems. [ABSTRACT FROM AUTHOR]

The properties of risk measures in the form of infimal convolution are analyzed. The dual representation of such measures, their subdifferential, extremum conditions, representation for optimization and use in constraints are described. The results of the study are demonstrated by examples of known risk measures of such structure. This allows systematization of the available results and facilitates a potential search for new variants of risk measures. [ABSTRACT FROM AUTHOR]

Properties of the apparatus of polyhedral coherent risk measures, its relationship with problems of robust and distributionally robust optimization, as well as its application under uncertainty are described. Problems of calculating robust structures of polyhedral coherent risk measures and their minimization, which are reduced to the corresponding linear programming problems, are considered. [ABSTRACT FROM AUTHOR]

Polyhedral coherent risk measures are extended to the case of imprecise scenario estimates of random variables. Optimization problems under uncertainty are considered that cover a wide class of stochastic programming and robust optimization problems. It is shown how they are reduced to linear programming problems in the linear case. Problems of portfolio optimization by the reward-to-risk ratio are considered. [ABSTRACT FROM AUTHOR]