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A Polyhedral Approximation Framework for Convex and Robust Distributed Optimization
, 2013
"... In this paper we consider a general problem setup for a wide class of convex and robust distributed optimization problems in peertopeer networks. In this setup convex constraint sets are distributed to the network processors who have to compute the optimizer of a linear cost function subject to ..."
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In this paper we consider a general problem setup for a wide class of convex and robust distributed optimization problems in peertopeer networks. In this setup convex constraint sets are distributed to the network processors who have to compute the optimizer of a linear cost function subject to the constraints. We propose a novel fully distributed algorithm, named cuttingplane consensus, to solve the problem, based on an outer polyhedral approximation of the constraint sets. Processors running the algorithm compute and exchange linear approximations of their locally feasible sets. Independently of the number of processors in the network, each processor stores only a small number of linear constraints, making the algorithm scalable to large networks. The cuttingplane consensus algorithm is presented and analyzed for the general framework. Specifically, we prove that all processors running the algorithm agree on an optimizer of the global problem, and that the algorithm is tolerant to node and link failures as long as network connectivity is preserved. Then, the cutting plane consensus algorithm is specified to three different classes of distributed optimization problems, namely (i) inequality constrained problems, (ii) robust optimization problems, and (iii) almost separable optimization problems with separable objective functions and coupling constraints. For each one of these problem classes we solve a concrete problem that can be expressed in that framework and present computational results. That is, we show how to solve: position estimation in wireless sensor networks, a distributed robust linear program and, a distributed microgrid control problem.
A Distributed Simplex Algorithm and the MultiAgent Assignment Problem
"... Abstract — In this paper we propose a novel distributed algorithm to solve degenerate linear programs on asynchronous networks. Namely, we propose a distributed version of the well known simplex algorithm. We prove its convergence to the global lexicographic minimum for possibly fully degenerate pro ..."
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Abstract — In this paper we propose a novel distributed algorithm to solve degenerate linear programs on asynchronous networks. Namely, we propose a distributed version of the well known simplex algorithm. We prove its convergence to the global lexicographic minimum for possibly fully degenerate problems and provide simulations supporting the conjecture that the completion time scales linearly with the diameter of the graph. The algorithm can be interpreted as a dual version of programs when the last is applied to linear programs. Finally, we study a multiagent task assignment problem and show that it can be solved by means of our distributed simplex algorithm. I.
Chapter 1 Network Abstract Linear Programming with Application to Cooperative Target Localization∗
"... Abstract We identify a novel class of distributed optimization problems, namely a networked version of abstract linear programming. For such problems we propose distributed algorithms for networks with various connectivity and/or memory constraints. Finally, we show how a suitable target localizati ..."
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Abstract We identify a novel class of distributed optimization problems, namely a networked version of abstract linear programming. For such problems we propose distributed algorithms for networks with various connectivity and/or memory constraints. Finally, we show how a suitable target localization problem can be tackled through appropriate linear programs. 1.1