Fixed Point Optimization Algorithms for Distributed Optimization in Networked Systems

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This paper considers a networked system with a finite number of users and deals with the problem of minimizing the sum of all users' objective functions over the intersection of all users' constraint sets, onto which the projection cannot be easily implemented. The main objective of this paper is to devise distributed optimization algorithms, which enable each user to find the solution of the problem without using other users' objective functions and constraint sets. To reach this goal, we first introduce easily implementable nonexpansive mappings of which the intersection of the fixed point sets is equal to the constraint set in the problem. We formulate the problem as a convex minimization problem over the intersection of the fixed point sets of the nonexpansive mappings. We then present an iterative algorithm, based on the conventional incremental subgradient methods which use the projection, for solving the problem. The algorithm can be implemented by using nonexpansive mappings other than the projection. We prove that the algorithm with slowly diminishing step-size sequences converges to a solution of the problem in the sense of weak topology of a Hilbert space. We also present a broadcast type of distributed optimization algorithm that weakly converges to a solution of the problem. Numerical examples for the bandwidth allocation demonstrate the convergence of these algorithms.