Smoothing D.C. Approximation To Chance Constraints Programming Based On Chen-Harker-kanzow-smale Smooth Plus Function

Many important practical problems can be formulated as probability constrained programs. Research on probability constraint optimization problem has great significance in theory and application. In this paper, a smoothing D.C. approximation is explored to solve the probability constraint optimization problem based on Chen-Harker-Kanzow-Smale(CHKS) smooth plus function. A smoothing approximation to probability constrained function is proposed and the corresponding smoothing D.C. approximation problem is established. It is proved that the approximating problem is equivalent to the original one under certain conditions. Moreover, corresponding sample average approximation problem is built. Sequential convex approximation(SCA) algorithm is implemented to solve the smoothing D.C. approximation problem. The main contents of this paper are summarized as follows.Chapter 1 reviews the research background of the theory and algorithm of chance constrained programs. Preliminaries involved are introduced.In chapter 2, a smoothing D.C. function based on CHKS smooth plus function is proposed. Properties of the smoothing D.C. function are analyzed and the corresponding smooth D.C. approximation problem is established. The equivalence between approximation problem and original one is proved under certain conditions. When the parameter is sufficiently small, feasible region, optimal solution set, optimal value, and KKT pairs set of approximation problem converge to the counterparts of the original problem respectively.Chapter 3 establishes the sample average approximation function of the probability constrained function. And corresponding sample average approximation problem is built. Optimal solution set and optimal value of sample average approximation problem converge to those of the approximation problem respectively with probability 1 as the sample size is large enough.In chapter 4, sequential convex approximation method is proposed to solve the smooth D.C. approximation problem. It introduces the sequential convex approximation(SCA) algorithm. Gradient-based Monte Carlo method is applied to solve a convex sub-problem in each iteration. It is shown that SCA algorithm has some desired convergence properties.In chapter 5, the SCA algorithm program is written in Matlab, in which function fmincon is called for solving convex sub-problem. And an example is computed by SCA algorithm. Numerical results reported shown that smoothing D.C. approximation proposed is feasible for solving probability constraint optimization problem.