An Extragradient Method For Variational Inequality Problems And Its Application To Optimal Control

As a branch of applied mathematics,variational inequality problems have been widely concerned for a long time.Because of its wide application in the fields of differential equations,optimal control,mathematical economy,countermeasure theory,and nonlinear optimization,how to effectively solve variational inequalities has always been a hot topic of research.In the first part,a subgradient extragradient method with inertial effects is proposed for solving variational inequalities.Then the convergence of the method is carried out,and the weak convergence of the sequence generated by the algorithm to the solution of variational inequalities is proved.Finally,numerical experiments are carried out to demonstrate the advantages of the proposed algorithm.In the second part,we mainly study the application of variational inequalities in constrained linear quadratic optimal control problems.By means of the first-order optimality condition,it is equivalently transformed into a monotone variational in-equality problem.The regularity of the constrained linear quadratic optimal control problem is studied by using the Tikhonov regularization method of variational in-equalities.It is proved that the solution of the perturbation problem converges to the minimum norm solution of the original problem.Finally,an extragradient method for solving stochastic linear quadratic optimal control problem is proposed,and numerical experiments are carried out.The applicability and validity of the algorithm are demonstrated by comparing the exact solution with the numerical solution.