Two Stochastic Approximation Projection Algorithms For Solving Stochastic Variational Inequality

Since the introduction of the variational inequality(?)problem,the research on its theory and application has made significant progress.There are already relatively complete set of theories and methods.However,there are uncertain factors in practical applications.The algorithm of ? problem is invalid.Therefore,it is necessary to study the random variational inequality(SVI)problem.This thesis mainly studies the stochastic approximation projection algorithm for solving SVI problems.Firstly,in the introduction part,the definitions of several types of monotonic functions,the definition of projection and its properties,the notation,the background of ? problems and the research status of projection algorithms,the research status of SVI problems and related algorithms are introduced.Then,the extragradient stochastic approximation projection algorithm for solving SVI problems is studied.According to the extragradient projection algorithm for solving classical ? problems,a modified extragradient stochastic approximation projection algorithm for solving SVI problems is given,which is referred to as MESA algorithm.Under appropriate assumptions,it is proved that the MESA algorithm converges with probability 1.The preliminary numerical experiments show that the MESA algorithm is effective.The MESA algorithm is a further extension of the existing extragradient stochastic approximation projection algorithm,and they can be obtained under weak assumptions.Finally,research on solving SVI problem is not feasible stochastic approximation projection algorithm,referred to as IPSA algorithm.The proposed algorithm can be regarded as an improvement of the extragradient projection algorithm.The IPSA algorithm only needs one projection in each iteration.A new direction and step size is used in the correction step compared to the general extragradient projection algorithm.The absence of the Litschitz constant.The variance of the stochastic error is reduced during the iterative process,and the line search of the dynamic sample is used to deal with the loss of the Lipschitz constant.Under the assumptions that it is weaker than pseudomonotone and monotone,the IPSA algorithm converges with probability 1,and analyzes the complexity and convergence rate of the IPSA algorithm.Preliminary numerical experiments show that the IPSA algorithm is effective.