ON THE SMOOTHNESS OF AN OBJECTIVE FUNCTION IN QUANTILE OPTIMIZATION PROBLEMS

The paper is concerned with the issue of differentiability of the quan tile function, the criterion that is commonly used in stochastic optim ization problems. Two cases are studied: when the quantile function ca n be represented as an implicit function using the probability functio n, and when, the quantile function can be approximated by the maximum function over some confidence set. In the second case, the quantile fu nction turns out to be differentiable, which is not always the case fo r the probability function.