Abstract. 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.

Keywords: polyhedral coherent risk measure, conditional value-at-risk, spectral coherent risk measure, imprecise estimate, linear programming, portfolio optimization, reward-to-risk ratio.