A smooth approximation approach for optimization with probabilistic constraints based on sigmoid function

Many practical problems, such as computer science, communications network, product design, system control, statistics and finance, etc.,can be formulated as a probabilistic constrained optimization problem (PCOP) which is challenging to solve since it is usually nonconvex and nonsmooth. Effective methods for the probabilistic constrained optimization problem mostly focus on approximation techniques, such as convex approximation, D.C. (difference of two convex functions) approximation, and so on. This paper aims at studying a smooth approximation approach. A smooth approximation to the probabilistic constraint function based on a sigmoid function is analyzed. Equivalence of PCOP and the corresponding approximation problem are shown under some appropriate assumptions. Sequential convex approximation (SCA) algorithm is implemented to solve the smooth approximation problem. Numerical results suggest that the smooth approximation approach proposed is effective for optimization problems with probabilistic constraints.