Log-sigmoid Approximation Of Chance Constrained Programs

Many important practical problems can be formulate as chance constrained programs,which are always non-convex and non-smooth.Effective methods for chance constrainedprograms mostly are convex approximation.In this paper,we propose to smooth the chanceconstrained and establish the associating smoothed approximation problems based onLog-Sigmoid function.Moreover,we use sequential convex approximation algorithm basedon Monte Carlo method and sample average approximation approach to solve the smoothedapproximation problem respectively. The main research results are as follows:1.Chapter2studies Log-Sigmoid approximation of the primal chance constrainedprograms.Firstly, the properties of the Log-Sigmoid function are analyzed.Secondly,Log-Sigmoid approximation function of the chance constrained function is constructed andcorresponding Log-Sigmoid approximation problem is established.We prove the Log-Sigmoidapproximation problem and the original problem are equivalent.Finally,we analysis theconvergence properties of the Log-Sigmoid approximation problem based on theories.Whenthe parameter is enough small,feasible region,optimal value,optimal solution set and KKTpairs set of the Log-Sigmoid approximation problem converge to the counterparts of theoriginal problem respectively.2.Chapter3investigates the approaches for solving Log-Sigmoid approximationproblem and discusses the corresponding convergence. Firstly,we study sequential convexapproximation method base on the Monte Carlo and prove the sequence generated by thealgorithm has desired convergence properties.Secondly, the sample average approximationfunction is constructed and the corresponding sample average approximation problem isestablished. It is shown that optimal value,optimal solution set of sample averageapproximation problem converge to the counterparts of the Log-Sigmoid approximationproblem respectively with probability1when the sample size is large enough.3. Chapter4builds the sequential convex approximation algorithm to solve theLog-Sigmoid approximation problem.We compile the computer procedure in Matlab, whichis implemented to an example.Numerical results reported verify the validity of the smoothapproximation approach based on Log-Sigmoid function.