Splitting Algorithms For Multistage Stochastic Variational Inequalities

The multistage stochastic variational inequality is transformed into a variational inequality with separable structure through introducing a new variable.Three classes of splitting algorithms which are originally used to solve deterministic variational inequalities,are used to solve the transformed separable structural variational inequalities,and obtain the efficient numerical algorithms for the original multistage stochastic variational inequality.Firstly,we give the definition of multistage stochastic variational inequalities in the general probability space by means of the probability theory and stochastic analysis.This definition is applicable for both discrete random variables and continuous random variables.Secondly,we prove the weak convergence of the ADMM algorithm for multistage stochastic variational inequalities under the condition of monotonicity and Lipschitz continuity in the general probability space.Thirdly,a prediction-correction ADMM algorithm for multistage stochastic variational inequalities is proposed.Under the condition of monotonicity and Lipschitz continuity,we prove the weak convergence of the prediction-correction ADMM algorithm.Compared with the ADMM algorithm,the prediction-correction ADMM algorithm is an explicit scheme algorithm.It is more effective for solving variational inequality problems with complex structures.Furthermore,an inexact parallel splitting algorithm for multistage stochastic variational inequalities is proposed.Under the condition of monotonicity and Lipschitz continuity,we prove the weak convergence of the inexact parallel splitting algorithm.Compared with the prediction-correction ADMM algorithm,the inexact parallel splitting algorithm is also an explicit algorithm,but the computation of the two projections is performed in parallel regardless of the order.Finally,a numerical example of those algorithms is given in the case of discrete probability spaces.The experimental results show that the prediction-correction ADMM algorithm and the inexact parallel splitting algorithm,as explicit algorithms,have higher computational efficiency than implicit algorithms.