This paper describes the software complex for the construction of a smooth approximation of the probability function and its derivatives. The structural parts of the complex, its functionality and mathematical background are described. The software complex constructs an approximation of the probability that some loss function does not exceed a certain level of loss. The program supports lossfunctions defined in a LaTeX format and contains predefined standard functions, variables and mathematical signs. The software supports different variable types, such as constants, control variables, stochastic variables with known distribution and parameters and samples of stochastic variables with unknown distribution. The software complex supports a variety of predefined random distributions and allows to tune the result by setting other service parameters. The implemented approximation is based on the replacement of the Heaviside function inside the probability function expression with the sigmoid function. Next, the approximated probability function and its derivatives are represented as volume integrals. These integrals can be calculated numerically using the Monte-Carlo method. This approach provides a relatively quick and universal method of approximate calculation of the probability function and its derivatives. The software complex has a graphical user interface and produces a graphical representation of approximated functions along with their points data. The program also supports the construction of the surface approximations for the case of the loss function having two control variables. Obtained graphical and point data can be used in the solution of stochastic programming problems with probability criteria. Examples using the software complex as a tool for analyzing stochastic programming problems are given.

In this paper, we provide an approximation method for probability function and its derivatives, which allows using the first order numerical algorithms in stochastic optimization problems with objectives of that type. The approximation is based on the replacement of the indicator function with a smooth differentiable approximation – the sigmoid function. We prove the convergence of the approximation to the original function and the convergence of their derivatives to the derivatives of the original ones. This approximation method is highly universal and can be applied in other problems besides stochastic optimization – the approximation of the kernel of the probability measure, considered in the present article as an example, and the confidence absorbing set approximations.

In this paper we study one of the possible variants of smooth approximation of probability criteria in stochastic programming problems. The research is applied to the optimization problems of the probability function and the quantile function for the loss functional depending on the control vector and one-dimensional absolutely continuous random variable. In this paper we study one of the possible variants of smooth approximation of probability criteria in stochastic programming problems. The research is applied to the optimization problems of the probability function and the quantile function 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 a 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 us 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 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. Next, approximate expressions for the derivatives of the probability function by the control vector and the parameter of the function are obtained, their convergence to the true derivatives is proved under a number of conditions for the loss functional. Using known relations between derivatives of probability functions and quantile functions, approximate expressions for derivatives of quantile function by control vector and by the level of probability are obtained. 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 in the case of a multidimensional random variable.