An extension of the quantile optimization problem with a loss function linear in random parameters

This paper studies the stochastic programming problem with a quantile criterion in the classical single-stage statement under the assumption that the loss function is linear in random parameters. An extension of this problem is the minimax one in which the inner maximum of the loss function is taken with respect to the realizations of the vector of random parameters over the kernel of its probability distribution, and the outer minimum is taken with respect to the optimized strategy over a given set of admissible strategies. The extension principle of optimization problems is used to establish the following result: under a sufficient condition in the form of a certain probabilistic constraint, the optimal solution of this minimax problem is also optimal in the original problem with the quantile criterion.