AN C-RELAXATION METHOD FOR SEPARABLE CONVEX COST NETWORK FLOW PROBLEMS1

We propose a new method for the solution of the single commodity, separable convex cost network flow problem. The method generalizes the e-relaxation method developed for linear cost problems, and reduces to that method when applied to linear cost problems. We show that the method terminates with a near optimal solution, and we provide an associated complexity analysis. We also present computational results showing that the method is much faster than earlier relaxation methods, particularly for ill-conditioned problems. 1 Research supported by NSF under Grant CCR-9103804 and Grant 9300494-DMI. 2 Dept. of Electrical Engineering and Computer Science, M.I.T., Rm. 35-210, Cambridge, Mass., 02139. Email: dimitrib@mit.edu and lcpolyme@lids.mit.edu 3 Dept. of Math., Univ. of Washington, Seattle, Wash., 98195. Email: tseng@math.washington.edu

[1]  D. Bertsekas.  An auction algorithm for the max-flow problem , 1995 .

[2]  Danna Zhou,et al.  d. , 1840, Microbial pathogenesis.

[3]  Jose A. Ventura,et al.  Computational development of a lagrangian dual approach for quadratic networks , 1991, Networks.

[4]  David L. Jensen,et al.  On the computational behavior of a polynomial-time network flow algorithm , 1992, Math. Program..

[5]  D. Bertsekas,et al.  Partially asynchronous, parallel algorithms for network flow and other problems , 1990 .

[6]  Andrew Vladislav Goldberg,et al.  Efficient graph algorithms for sequential and parallel computers , 1987 .

[7]  S. Thomas McCormick,et al.  Polynomial Methods for Separable Convex Optimization in Unimodular Linear Spaces with Applications , 1997, SIAM J. Comput..

[8]  Pasumarti V. Kamesam,et al.  Multipoint methods for separable nonlinear networks , 1984 .

[9]  Dimitri P. Bertsekas,et al.  A forward/reverse auction algorithm for asymmetric assignment problems , 1992, Comput. Optim. Appl..

[10]  Dimitri P. Bertsekas,et al.  Distributed Asynchronous Relaxation Methods for Linear Network Flow Problems , 1987 .

[11]  Dimitri P. Bertsekas,et al.  RELAX-IV : a faster version of the RELAX code for solving minimum cost flow problems , 1994 .

[12]  R. Meyer.  Two-Segment Separable Programming , 1979 .

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

[14]  T. B. Boffey.  Linear Network Optimization: Algorithms and Codes , 1994 .

[15]  Dimitri P. Bertsekas,et al.  Parallel Shortest Path Auction Algorithms , 1994, Parallel Comput..

[16]  D. Bertsekas,et al.  Distributed asynchronous relaxation methods for convex network flow problems , 1987 .

[17]  Dimitri P. Bertsekas,et al.  A generic auction algorithm for the minimum cost network flow problem , 1993, Comput. Optim. Appl..

[18]  Paul Tseng,et al.  Relaxation methods for monotropic programming problems , 1986 .

[19]  D. Bertsekas,et al.  Relaxation methods for network flow problems with convex arc costs , 1987 .

[20]  Eric V. Denardo,et al.  Flows in Networks , 2011 .

[21]  Stavros A. Zenios,et al.  On the Massively Parallel Solution of Linear Network Flow Problems , 1991, Network Flows And Matching.

[22]  Paul Tseng,et al.  Relaxation Methods for Minimum Cost Ordinary and Generalized Network Flow Problems , 1988, Oper. Res..

[23]  W. Hager.  The Dual Active Set Algorithm and the Iterative Solution of Linear Programs , 2003 .

[24]  K. Fernow.  New York , 1896, American Potato Journal.

[25]  D. Bertsekas.  Distributed relaxation methods for linear network flow problems , 1986, 1986 25th IEEE Conference on Decision and Control.

[26]  A. Weintraub.  A Primal Algorithm to Solve Network Flow Problems with Convex Costs , 1974 .

[27]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[28]  William W. Hager,et al.  Application of the dual active set algorithm to quadratic network optimization , 1993, Comput. Optim. Appl..

[29]  Dimitri P. Bertsekas,et al.  Dual coordinate step methods for linear network flow problems , 1988, Math. Program..

[30]  Richard M. Karp,et al.  Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems , 1972, Combinatorial Optimization.

[31]  Dimitri P. Bertsekas,et al.  A unified framework for primal-dual methods in minimum cost network flow problems , 1985, Math. Program..

[32]  D. Klingman,et al.  NETGEN - A Program for Generating Large Scale (Un) Capacitated Assignment, Transportation, and Minim , 1974 .

[33]  Andrew V. Goldberg,et al.  Solving minimum-cost flow problems by successive approximation , 1987, STOC.