Stochastic quasi-gradient algorithms for maximization of the probability function. A new formula for the gradient of the probability function

New methods for constructing statistical estimates of the gradient of the probability function (stochastic quasi-gradient) are developed for the problems where the probability function is defined by a nonlinear system of inequalities with discontinuous, Lipschitz, and smooth functions. These methods are based on the representations of the probability function as an expectation of discontinuous, Lipschitz, and smooth functions. A new formula for the gradient of the probability function is obtained in the form of a volume integral over the unit simplex. The methods proposed are utilized in a stochastic quasi-gradient algorithm for maximization of the probability function.