PROBLEM1 minimizes Kantorovich-Rubinstein distance and finds the best approximating discrete distribution with a fixed number of atoms and variable probabilities and locations of the atoms. We considered two cases: without and with initial approximating distribution (initial point in the optimization problem).

PROBLEM 2 minimizes Average Kolmogorov-Smirnov distance and finds an approximating distribution with fixed locations and variable probabilities of the atoms. In this case, locations of atoms of the approximating distribution coincide with the location of atoms of the target distribution. To constraint the number of atoms with nonzero probabilities in the approximating distribution, we have imposed a cardinality constraint. We considered two options for the cardinality constraint: 1) “linearize=1”; 2) “mip=1”. These two options initiate different optimization approaches with cardinality constraints.

PROBLEM 3 and PROBLEM 4 minimize Average Kolmogorov-Smirnov distance and find an approximating distribution with fixed locations and variable probabilities of atoms. The number of atoms of the approximating distribution is 4 times smaller than the number of atoms of the target distribution. To assure that the right tail of the approximating distribution correctly matches the right tail of the target distribution we have imposed two CVaR constraints. We want to minimize distance between distributions and match two CVaRs of approximating and target distributions. We have considered two cases: 1) Problem 3: CVaRs of approximating distribution are larger than CVaRs of target distribution (in this case tails of approximating distribution are heavier than tail of target distribution). This problem is convex and optimality of the solution in guaranteed; 2) Problem 4: CVaRs of approximating distribution are equal to CVaRs of target distribution. This problem may have multiple extremums and solver finds some extremum(may be a local one).

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