The Analysis And Algorithms For Nonmonotone Stochastic Variational Inequalities: Single-stage To Multistage

Recently,the study of stochastic variational inequalities(SVIs)has attracted a great deal of attention in the optimization field.SVIs are closely related to the theory of stochastic optimization and play an important role in real world applications including economy,engineering and management.There are fruitful results about the theory and algorithms of monotone SVIs,while there are few results related to nonmonotone SVIs.Especially,the study of nonmonotone multistage SVIs is at the beginning.In this dissertation,we will focus on the theory and algorithms of nonmonotone SVIs.Firstly,we study the single-stage stochastic R0 matrix linear complementarity problems.We propose the Fischer-Burmeister(FB)function-based expected residual minimization(ERM)model,prove that the solution set of resultant optimization problem is nonempty and bounded if and only if the involved matrix is a stochastic R0 matrix,and show that the objective function is continuously differentiable in which indicate that the FB function-based ERM model is better than the existed min function-based ERM model.Secondly,we investigate the multistage pseudomonotone SVIs for the first time.Based on the constructed isomorphism between SVIs and deterministic variational inequalities,we establish theoretical results on the existence,convexity,boundedness and compactness of the solution set,propose the elicited progressive hedging algorithm(PHA)for solving pseudomonotone SVIs,and provide some sufficient conditions on the elicitability of pseudomonotone SVIs required by the convergence of the elicited PHA.Numerical results are presented to show the efficiency of the elicited PHA.Finally,we study the algorithms for the general nonmonotone SVIs.We introduce the SVIs satisfying locally elicitable maximal monotonicity(LEMM),and propose the localized PHA for solving this kind of SVIs for the first time.We prove the local convergence under LEMM condition,and provide some conditions for the linear convergence.Some numerical experiments indicate that the localized PHA is efficient.