Study On Theory And Algorithms For Stochastic Mathematical Programming With Equilibrium Constraints

Equilibrium problems have many important applications in operations research,com-puter science,system science,engineering technology,transportation,economics and man-agement ect.In the last 20 years of the twentieth century,great attention has been paid by many scholars to this direction.Because of the objective reflection of the random factors in reality,the stochastic programming problem,especially the stochastic math-ematical programming with equilibrium constraints is a hot research topic in the field of optimization.This dissertation focuses on the theory and algorithms for stochastic mathematical programming with equilibrium constraints from three aspects:convergence analysis of a algorithm for a stochastic mathematical programming with complementarity constraints,consistency analysis of stationary points for a stochastic bilevel programming and consistency analysis of stability of a parametric stochastic complementarity problem.The main results of this dissertation can be summarized as follows:1.In Chapter 3,a regularization sample average approximation scheme for solving a stochastic programming with complementarity constraints is proposed and the con-vergence theory of this method is established.It is demonstrated that,under suit-able constraint qualifications,the sequence of optimal solutions to the regularization sample average approximation converges to a optimal solution of original problem with probability one when the number of the samples tends to infinity.In what fol-lows,it is proved that under the regular condition of original problem,the sequence of stationary points of the regularization sample average approximation converges to a stationary point of original problem with probability one when the number of the samples tends to infinity.Moreover,the exponential convergence rate of the sequence of stationary points of the regularization sample average approximation to the corresponding stationary point of original problem is established under the strong second order sufficient conditions of original problem.Finally,the efficiency of regularization sample average approximation method is verified by some numer-ical examples.2.Chapter 4 focuses on the consistency theory of global optimal solutions of sample aver-age approximation stochastic bilevel programming,whose lower problem is a second order cone programming.At first,the concrete formula with the equality type of the coderivative of solution mapping to a second order cone constrained parametric variational inequalities is established under the constraint nondegeneracy condition and the strict complementarity condition.And then,on the basis of the equality type formula,a necessary and sufficient condition for the global optimal solution of a bilevel programming,whose lower problem is a second order cone programming,is obtained.At last,under some regular conditions,the sequence of global optimal solutions of the sample average approximation problems of a stochastic bilevel prob-lem with the second order cone programming lower problem converges to a global optimal solution of original problem with probability one when the number of the samples tends to infinity.3.In Chapter 5,the consistency theory of the Aubin property of sample average approx-imation solution mapping to a parametric stochastic complementarity problem is established.At first,under some constraint qualifications,the upper inclusion for-mula of coderivative of solution mapping to a parametric stochastic complementarity problem is established.In what follows,utilizing the concept of cosmic deviation in variational analysis,sufficient conditions ensuring the Aubin property of solution mapping of the sample average approximation parametric stochastic complemen-tarity problem are obtained.Finally,the results are applied to a concrete problem and the consistency of the Aubin property of solution mapping to this problem is obtained.