A Smoothing Method to Probabilistic Constrained Programs Based on Logarithm-exponential Function
|School||Liaoning Normal University|
|Course||Operational Research and Cybernetics|
|Keywords||Probabilistic Constraint Smoothing Function Convergence Analysis Sequential Convex Approximation Sample Average Approximation|
Stochastic optimization with probabilistic constraints has been widely used in manypractical problem, such as supply chain management, water resources management, riskoptimization, etc. Probabilistic constrained program has both important theoreticalsignificance and practical values. It is one of the hot topics in the field of stochasticoptimization. Because probability constraint functions are usually non-convex andnon-smooth, which faces challenges for solving. Effective methods mostly focus on convexapproximation technique. This thesis aims to studying a smoothing method to probabilisticconstrained programs based on Logarithm-Exponential (Log-Exp) function. It establishedcorresponding smooth approximation problem, which is solved by sequential convexapproximation and sample average approximation approach. The main research contents ofthis paper are as follows:Chapter1reviews the research background and lists basic knowledge of probabilityinvolved in research.Chapter2focuses on a smooth approximation to the probabilistic constrainedoptimization problem based on Log-Exp function. Firstly, D.C. function is introduced.Secondly, it discusses Log-Exp function and its properties. Thirdly, probabilistic constraintsare smoothed and corresponding approximation problem is established. It proves theequivalence of approximation problem and the original problem. Finally, It carries on theconvergence analysis for approximation problem based on theories involved in probabilitytheory and vaviational analysis theory. Under certain conditions, feasible region, optimalvalue, optimal solution set and KKT pairs set of approximation problem converge to thecounterparts of the original problem respectively when the parameter is sufficiently small.Chapter3studies the methods for solving smooth approximation problem. Firstly, itproposes sequential convex approximation method and proves the sequence generated by thealgorithm has good convergence properties. Secondly, sample average approximaton problemis established. It proves that sample average approximation problem is equivalent to thesmooth approximation problem with probability1when the sample size is large enough.Under certain conditions, optimal value, optimal solution set of sample averageapproximation problem converge to the counterparts of the smooth approximation problemrespectively with probability1.