Parallel Optimization Algorithm for Smooth Convex Optimization over Fixed Point Sets of Quasi-nonexpansive Mappings

Smooth convex optimization problems are solved over fixed point sets of quasi-nonexpansive mappings by using a distributed optimization technique. This is done for a networked system with an operator, who manages the system, and a finite number of users, by solving the problem of minimizing the sum of the operator’s and users’ differentiable, convex objective functions over the intersection of the operator’s and users’ fixed point sets of quasi-nonexpansive mappings. Under the assumption that the operator can communicate with all users, a parallel optimization algorithm can be devised that enables the operator to find a solution to the problem without using all user objective functions and quasi-nonexpansive mappings. This algorithm does not use proximity operators, in contrast to conventional parallel proximal algorithms. Moreover, it can optimize over fixed point sets of quasi-nonexpansive mappings, in contrast to conventional fixed point algorithms. Investigation of the algorithm’s convergence properties for a constant step-size rule reveals that, with a small constant step size, it approximates the solution to the problem. Consideration of the case in which the step-size sequence is diminishing demonstrates that the algorithm converges to the problem solution. Application of the algorithm to network bandwidth allocation based on an operational policy is shown to make the network more stable and reliable.