DISTRIBUTED ITERATIVE METHODS FOR SOLVING NONMONOTONE VARIATIONAL INEQUALITY OVER THE INTERSECTION OF FIXED POINT SETS OF NONEXPANSIVE MAPPINGS

We consider a networked system which consists of a finite number of users and discuss a non-monotone variational inequality with the gradient of the sum of all users' nonconvex objective functions over the intersection of all users' constraint sets. Centralized iterative methods, which require us to use the explicit forms of all users' objective functions and constraint sets, have been proposed to solve the variational inequality. However, it would be difficult to apply them to the variational inequality in this system because none of the users in the system can know the explicit forms of all users' objective functions and constraint sets. In this paper, we translate the variational inequality into a nonmonotone variational inequality over the intersection of the fixed point sets of certain nonexpansive mappings and devise distributed iterative methods, which enable each user to solve the variational inequality without using the explicit forms of other users' objective functions and constraint sets, based on fixed point theory for nonexpansive mappings. We also present convergence analyses for them and provide numerical examples for the bandwidth allocation. The analyses and numerical examples suggest that distributed iterative methods with slowly diminishing step-size sequences converge to a solution to the variational inequality.