On smooth approximation of probabilistic criteria in stochastic programming problems

One of the possible variants of smooth approximation of probability criteria in stochastic programming problems allowing to obtain estimates of the probability function gradient and the quantile function gradient in the form of a volume integral is considered. The research is applied to problems of probability function maximization and quantile function minimization for the loss functional depending on the control vector and one-dimensional absolutely continuous random variable.
The main idea of the approximation is to replace the discontinuous Heaviside function in the integral representation of the probability function with a smooth function having such properties as continuity, smoothness, and easily computable derivatives. An example of such function is the distribution function of a random variable distributed according to the logistic law with zero mean and finite dispersion, which is a sigmoid. The value inversely proportional to the root of the variance is a parameter that provides the proximity of the original function and its approximation. This replacement allows to obtain a smooth approximation of the probability function, and for this approximation derivatives by the control vector and by other parameters of the problem can be easily found.
The main result of the article is the obtained expressions for approximation of the probability function derivatives by the control vector and by the acceptable loss level, as well as expressions for approximation of the quantile function gradient in the form of volume integrals. The article proves the convergence of the probability function approximation obtained by replacing the Heaviside function with the sigmoidal function to the original probability function, and the error estimate of such approximation is obtained. The convergence of the approximation of probability function derivatives to the true derivatives under a number of conditions on the loss functional is also proved.
Examples are considered to demonstrate the possibility of applying the proposed estimates to the solution of stochastic programming problems with criteria in the form of a probability function and a quantile function, including the case of a multidimensional random variable