Abstract
In this paper, we consider a stochastic optimization problem with a convex piecewise linear (polyhedral) loss function, polyhedral constraints, and continuous random vector distribution. The objective is to maximize the probability that the loss function value does not exceed the specified level when constraints are satisfied. We use an approximation of polyhedral function and probability criterion, replacing maximum with smooth maximum transform and Heaviside function with sigmoid function. This replacement leads to the approximation of their gradients. Next, the problem solved using modified gradient descent. The accuracy and effectiveness of this method are shown in two examples. Also, we discuss some aspects of such approximations in the case of discrete random distribution.
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