A Smooth Approximation To Chance Constrained Programming Based On Sigmoid Function

Stochastic optimization with probabilistic constraints is a research topic which has bothimportant theoretical significance and practical values in the area of stochastic optimization.Many important practical problems can be formulated as probabilistic constrained programs,which usually are non-convex and non-smooth. Effective methods for stochastic optimizationproblems with probabilitic constraints mostly focus on convex approximation techniques.Thisthesis aims at studying smooth approximation to stochastic programs with probabilisticconstraints based on Sigmoid function. It establishes corresponding smooth approximationproblem, which is solved by sample average approximation approach. The main researchcontents of this paper are as follows:Chapter1reviews the research background and lists basic knowledge of probabilityinvolved in research.Chapter2introduces in brief two effective methods for solving probabilisticconstrained optimization: CVaR and D.C. approximations.It analysis the characteristics oftheir own by means of image.Chapter3establishes a smooth approximation to the probabilistic constrainedoptimization problem based on Sigmoid function. Firstly, properties of Sigmoid function areanalyzed.Secondly, probabilistic constraints are smoothed and corresponding approximationproblem is established. It proves the equivalence of approximation problem and the originalproblem. Thirdly, It carries on the convergence analysis for approximation problem based ontheories involved in probability theory and vaviational analysis theory. Under certainconditions, feasible region, optimal value, optimal solution set and multiplier set ofapproximation problem converge to the counterparts of the original problem respectivelywhen the parameter is sufficiently large.Chapter4proposes sample average approximaton approach for solving smoothapproximation problem. Firstly, sample average approximation problem is established. Itproves that sample average approximation problem is equivalent to the smooth approximationproblem with probability1when the sample size is large enough.Secondly, Under certainconditions, optimal value, optimal solution set of sample average approximation problemconverge to the counterparts of the smooth approximation problem respectively withprobability1.