Relaxation methods for network flow problems with convex arc costs

We consider the standard single commodity network flow problem with both linear and strictly convex possibly nondifferentiable arc costs. For the case where all arc costs are strictly convex we study the convergence of a dual Gauss-Seide! type relaxation method that is well suited for parallel computation. We then extend this method to the case where some of the arc costs are linear. As a special case we recover a relaxation method for the linear minimum cost network flow problem proposed in Bertsekas (1) and Bertsekas and Tseng (2).

[1]  D. R. Fulkerson,et al.  Flows in Networks. , 1964 .

[2]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[3]  E. Polak,et al.  Computational methods in optimization : a unified approach , 1972 .

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

[5]  D. Bertsekas,et al.  A DESCENT NUMERICAL METHOD FOR OPTIMIZATION PROBLEMS WITH NONDIFFERENTIABLE COST FUNCTIONALS , 1973 .

[6]  M. J. D. Powell,et al.  On search directions for minimization algorithms , 1973, Math. Program..

[7]  R. Sargent,et al.  On the convergence of sequential minimization algorithms , 1973 .

[8]  Darwin Klingman,et al.  NETGEN: A Program for Generating Large Scale Capacitated Assignment, Transportation, and Minimum Cost Flow Network Problems , 1974 .

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

[10]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .

[11]  R. Cottle,et al.  On the convergence of a block successive over-relaxation method for a class of linear complementarity problems , 1982 .

[12]  D. Bertsekas,et al.  Relaxation methods for minimum cost network flow problems , 1983 .

[13]  David G. Luenberger,et al.  Linear and nonlinear programming , 1984 .

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

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

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

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

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