A method for solving quantile optimization problems with a bilinear loss function

We propose a numerical method for solving quantile optimization problems with a bilinear loss function based on approximating the kernel of the probability measure in the space of realizations of the random parameters vector with a convex polyhedron. The original problem reduces to a linear programming problem with a large number of constraints. We present our approach in two modifications: for the case when we know the distribution of random parameters and for the case when we only have a sample from the distribution law. The operation of the proposed approach is illustrated with numerical solutions of portfolio selection.