A distributed simplex algorithm for degenerate linear programs and multi-agent assignments

In this paper we propose a novel distributed algorithm to solve degenerate linear programs on asynchronous peer-to-peer networks with distributed information structures. We propose a distributed version of the well-known simplex algorithm for general degenerate linear programs. A network of agents, running our algorithm, will agree on a common optimal solution, even if the optimal solution is not unique, or will determine infeasibility or unboundedness of the problem. We establish how the multi-agent assignment problem can be efficiently solved by means of our distributed simplex algorithm. We provide simulations supporting the conjecture that the completion time scales linearly with the diameter of the communication graph.

[1]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[2]  Colin Neil Jones,et al.  Lexicographic perturbation for multiparametric linear programming with applications to control , 2007, Autom..

[3]  Francesco Bullo,et al.  Monotonic Target Assignment for Robotic Networks , 2009, IEEE Transactions on Automatic Control.

[4]  Jason R. Marden,et al.  Autonomous Vehicle-Target Assignment: A Game-Theoretical Formulation , 2007 .

[5]  Haimonti Dutta,et al.  Distributed Linear Programming and Resource Management for Data Mining in Distributed Environments , 2008, 2008 IEEE International Conference on Data Mining Workshops.

[6]  Sandro Zampieri,et al.  Distributed Quasi-Newton Method and its Application to the Optimal Reactive Power Flow Problem , 2010 .

[7]  Sukhamay Kundu.  A Distributed O(|E|) Algorithm for Optimal Link-Reversal , 2009, ICDCN.

[8]  Dimitri P. Bertsekas,et al.  Network optimization : continuous and discrete models , 1998 .

[9]  Kevin M. Passino,et al.  Distributed Task Assignment for Mobile Agents , 2007, IEEE Transactions on Automatic Control.

[10]  G. Dantzig,et al.  THE DECOMPOSITION ALGORITHM FOR LINEAR PROGRAMS , 1961 .

[11]  Dimitri P. Bertsekas,et al.  Parallel synchronous and asynchronous implementations of the auction algorithm , 1991, Parallel Comput..

[12]  Giuseppe Notarstefano,et al.  Distributed Abstract Optimization via Constraints Consensus: Theory and Applications , 2011, IEEE Transactions on Automatic Control.

[13]  Emo Welzl,et al.  Linear Programming - Randomization and Abstract Frameworks , 1996, STACS.

[14]  D. Bertsekas.  The auction algorithm: A distributed relaxation method for the assignment problem , 1988 .

[15]  Sonia Martínez,et al.  On Distributed Convex Optimization Under Inequality and Equality Constraints , 2010, IEEE Transactions on Automatic Control.

[16]  Jens Vygen,et al.  The Book Review Column1 , 2020, SIGACT News.

[17]  D.A. Castanon,et al.  Distributed algorithms for dynamic reassignment , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[18]  George J. Pappas,et al.  A distributed auction algorithm for the assignment problem , 2008, 2008 47th IEEE Conference on Decision and Control.

[19]  Hans-Jakob Lüthi,et al.  The existence of a short sequence of admissible pivots to an optimal basis in LP and LCP , 1997 .

[20]  Han-Lim Choi,et al.  Consensus-Based Auction Approaches for Decentralized Task Assignment , 2008 .

[21]  Francesco Bullo,et al.  Distributed Control of Robotic Networks , 2009 .

[22]  C. B. Stunkel,et al.  Hypercube implementation of the simplex algorithm , 1989, C3P.

[23]  H. Kuhn.  The Hungarian method for the assignment problem , 1955 .

[24]  Gavriel Yarmish,et al.  A distributed, scaleable simplex method , 2008, The Journal of Supercomputing.

[25]  K. I. M. McKinnon,et al.  ASYNPLEX, an asynchronous parallelrevised simplex algorithm , 1998, Ann. Oper. Res..

[26]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

[27]  Rainer E. Burkard,et al.  Selected topics on assignment problems , 2002, Discret. Appl. Math..

[28]  E. Polak.  Introduction to linear and nonlinear programming , 1973 .

[29]  Asuman E. Ozdaglar,et al.  Constrained Consensus and Optimization in Multi-Agent Networks , 2008, IEEE Transactions on Automatic Control.

[30]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.