A Smooth Function Method For Chance Constrained Optimization Problem

Chance constrained programming was proposed by A.Charnes and W.W.Cooper in1959.Taking into account the decisions made in the adverse circumstances may not satisfy the constraint conditions,thus it is an optimal theory in a certain sense of probability. As an important branch of stochastic programming, chance constrained optimization has a high application value in various fields, so it gradually attracts the attentions from scholars in recent years.Although there are lots of research results in terms of solving chance constrained programming, there are still many problems especially the calculation method for the problems have not yet been resolved satisfactorily. Inspired by a recent work by L.Jeff Hong and Liwei Zhang which proposes to construct a non-smooth DC(difference of two convex)function and use a sequence of convex approximation algorithm to solve the joint chance constrained optimization problem, we consider to improve this method through construct a smooth DC function and apply it to the single chance constrained optimization problem.In this paper, we first use a smooth DC function which with a parameter to ap-proximate the constraint of the chance constrained optimization problem, then,we get a smooth approximation of the probability constraint problem; We show that when the parameter tends to0, the Karush-Kuhn-Tuck(KKT) points of DC problem converge to a KKT point of the original problem under certain assumptions; We propose to solve the smooth DC optimization problem by a sequence of convex approximations(SCA); Further more,for each convex sub-problem, as the function of the problem is defined by the math-ematical expectation, we propose to use a sample average approximation (SAA) method to solve the sequence of convex approximations, and the illustrative numerical results are given.