Smoothing D.C. Approximation To Probability Constrained Programming Based On Pinar-Zenios Smooth Plus Function

Probabilistic constrained optimization problem has been widely used in various fields in our real life,which has become a hot studying focus in recent years.The mathematical model of many practical problems is a probabilistic constrained optimization model.There are three main difficulties in solving the model.First,the probability constraint function p?x?is usually non-convex.Second,p?x?has no closed form.Third,p?x?is not differentiable.The scholars at home and abroad have done a lot of researches,and given some effective methods to solve the probabilistic constrained optimization problem,such as quadratic approximation,CVaR approximation,Bernstein approximation,second order cone programming approximation,D.C.approximation,smooth approximation,and so on.In this paper,for the non-differentiability of p?x?,based on Pinar-Zenios smoothing plus function,we establish the equivalent smooth D.C.approximation problem,discuss the sequential convex approximation?SCA?algorithm for solving the smooth D.C.approximation problem and construct the sample average approximation problem for the smooth D.C.approximation problem,we also carry out the convergence analysis.The main contents of this paper are summarized as follows.Chapter 1 We introduce the developed background of the probabilistic constrained optimization problem,and give the preliminary knowledge of the basic concepts and basic theorems.Chapter 2 Based on Pinar-Zenios smoothing plus function,we construct a smooth approximation function??z,t?of the characteristic function1_?0,+???z?,and the equivalent smooth D.C.approximation problem???is presented.We establish the?-approximation problem?P_??for the smooth approximation problem???,we prove that the approximation problem?P_??is equivalent to the original probabilistic constrained optimization problem,and the convergence analysis is also carried out.Chapter 3 We discuss the sequential convex approximation?SCA?method for solving the smooth D.C.approximation problem?P_??.First,we introduce SCA algorithm,and analyze the convergence properties of the algorithm.Second,we discuss the SCA method for solving the smooth D.C.approximation problem?P_??,give an effective method to determine the initial solution in the implementation process of SCA algorithm.For the convex sub-problems,we establish the sample average approximation problem,and introduce that it can be directly solved by a gradient-based Monte Carlo method.