This paper considers a networked system consisting of an operator, which manages the system, and a finite number of subnetworks with all users, and studies the problem of minimizing the sum of the operator's and a...
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This paper considers a networked system consisting of an operator, which manages the system, and a finite number of subnetworks with all users, and studies the problem of minimizing the sum of the operator's and all users' objective functions over the intersection of the operator's and all users' constraint sets. When users in each subnetwork can communicate with each other, they can implement an incremental subgradient method that uses the transmitted information from their neighbor users. Since the operator can communicate with users in the subnetworks, it can implement a broadcast distributed algorithm that uses all available information in the subnetworks. We present an iterative method combining broadcast and incremental distributed optimizationalgorithms. Our method has faster convergence and a wider range of application compared with conventional distributed algorithms. We also prove that under certain assumptions our method converges to the solution to the problem in the sense of the strong topology of a Hilbert space. Moreover, we numerically compare our method with the conventional distributed algorithms in the case of a data storage system. The numerical results demonstrate the effectiveness and fast convergence of our method.
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, ont...
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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 optimizationalgorithms, 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 optimizationalgorithm that weakly converges to a solution of the problem. Numerical examples for the bandwidth allocation demonstrate the convergence of these algorithms.
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