A truncated primal‐infeasible dual‐feasible network interior point method

In this paper, we introduce the truncated primal-infeasible dual-feasible interior point algorithm for linear programming and describe an implementation of this algorithm for solving the minimum-cost network flow problem. In each iteration, the linear system that determines the search direction is computed inexactly, and the norm of the resulting residual vector is used in the stopping criteria of the iterative solver employed for the solution of the system. In the implementation, a preconditioned conjugate gradient method is used as the iterative solver. The details of the implementation are described and the code PDNET is tested on a large set of standard minimum-cost network flow test problems. Computational results indicate that the implementation is competitive with state-of-the-art network flow codes. © 2000 John Wiley & Sons, Inc.

[1]  J. Kruskal.  On the shortest spanning subtree of a graph and the traveling salesman problem , 1956 .

[2]  R. Prim.  Shortest connection networks and some generalizations , 1957 .

[3]  D. R. Fulkerson,et al.  An Out-of-Kilter Method for Minimal-Cost Flow Problems , 1960 .

[4]  Garth P. McCormick,et al.  A Guide to SUMT-Version 4: The Computer Program Implementing the Sequential Unconstrained Minimization Technique for Nonlinear Programming, , 1965 .

[5]  Ellis L Johnson,et al.  PROGRAMMING IN NETWORKS AND GRAPHS , 1965 .

[6]  Gerald G. Brown,et al.  Design and Implementation of Large-Scale Primal Transshipment Algorithms , 1976 .

[7]  J. Meijerink,et al.  An iterative solution method for linear systems of which the coefficient matrix is a symmetric -matrix , 1977 .

[8]  John M. Mulvey,et al.  Testing of a large-scale network optimization program , 1978, Math. Program..

[9]  R. V. Helgason,et al.  Algorithms for network programming , 1980 .

[10]  F. Glover,et al.  The simplex SON algorithm for LP/embedded network problems , 1981 .

[11]  R. Dembo,et al.  INEXACT NEWTON METHODS , 1982 .

[12]  Trond Steihaug,et al.  Truncated-newtono algorithms for large-scale unconstrained optimization , 1983, Math. Program..

[13]  Robert E. Tarjan,et al.  Data structures and network algorithms , 1983, CBMS-NSF regional conference series in applied mathematics.

[14]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, Comb..

[15]  M. D. Grigoriadis,et al.  An efficient implementation of the network simplex method , 1986 .

[16]  Kunio Tanabe,et al.  Centered newton method for mathematical programming , 1988 .

[17]  Donald Goldfarb,et al.  A computational comparison of the dinic and network simplex methods for maximum flow , 1988 .

[18]  M. Kojima,et al.  A primal-dual interior point algorithm for linear programming , 1988 .

[19]  A. Goldberg,et al.  A new approach to the maximum-flow problem , 1988, JACM.

[20]  Bruno Simeone.  Fortran codes for network optimization , 1988 .

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

[22]  R. C. Monteiro,et al.  Interior path following primal-dual algorithms , 1988 .

[23]  Clyde L. Monma,et al.  An Implementation of a Primal-Dual Interior Point Method for Linear Programming , 1989, INFORMS J. Comput..

[24]  N. Megiddo.  Pathways to the optimal set in linear programming , 1989 .

[25]  Renato D. C. Monteiro,et al.  Interior path following primal-dual algorithms. part I: Linear programming , 1989, Math. Program..

[26]  Pravin M. Vaidya,et al.  An algorithm for linear programming which requires O(((m+n)n2+(m+n)1.5n)L) arithmetic operations , 1990, Math. Program..

[27]  Sanjay Mehrotra,et al.  A computational comparison of the network simplex method with the dual affine scaling method , 1991 .

[28]  Dimitri P. Bertsekas,et al.  Linear network optimization - algorithms and codes , 1991 .

[29]  K. G. Ramakrishnan,et al.  An Approximate Dual Projective Algorithm for Solving Assignment Problems , 1991, Network Flows And Matching.

[30]  Pravin M. Vaidya,et al.  A Fast Implementation of a Path-Following Algorithm for Maximizing a Linear Function over a Network Polytope , 1991, Network Flows And Matching.

[31]  K. G. Ramakrishnan,et al.  Computational results of an interior point algorithm for large scale linear programming , 1991, Math. Program..

[32]  Yinyu Ye,et al.  On the finite convergence of interior-point algorithms for linear programming , 1992, Math. Program..

[33]  Sanjay Mehrotra,et al.  On the Implementation of a Primal-Dual Interior Point Method , 1992, SIAM J. Optim..

[34]  Shinji Mizuno,et al.  A primal—dual infeasible-interior-point algorithm for linear programming , 1993, Math. Program..

[35]  David S. Johnson,et al.  Network Flows and Matching: First DIMACS Implementation Challenge , 1993 .

[36]  M. Resende,et al.  An Implementation of the Dual Affine Scaling Algorithm for Minimum-Cost Flow on Bipartite Uncapacitated Networks , 1993, SIAM J. Optim..

[37]  Andrew V. Goldberg,et al.  An efficient implementation of a scaling minimum-cost flow algorithm , 1993, IPCO.

[38]  Yinyu Ye,et al.  A Short-Cut Potential Reduction Algorithm for Linear Programming , 1993 .

[39]  Sanjay Mehrotra,et al.  Finding an interior point in the optimal face of linear programs , 1993, Math. Program..

[40]  Takashi Tsuchiya,et al.  A simplified global convergence proof of the affine scaling algorithm , 1993, Ann. Oper. Res..

[41]  Yinyu Ye,et al.  Convergence behavior of interior-point algorithms , 1993, Math. Program..

[42]  Roy E. Marsten,et al.  Feature Article - Interior Point Methods for Linear Programming: Computational State of the Art , 1994, INFORMS J. Comput..

[43]  M. Resende,et al.  Identifying the Optimal Face of a Network Linear Program with a Globally Convergent Interior Point Method , 1994 .

[44]  Roy E. Marsten,et al.  Rejoinder - The Last Word on Interior Point Methods for Linear Programming - For Now , 1994, INFORMS J. Comput..

[45]  Yin Zhang,et al.  A Study of Indicators for Identifying Zero Variables in Interior-Point Methods , 1994, SIAM Rev..

[46]  Masakazu Muramatsu,et al.  Global Convergence of a Long-Step Affine Scaling Algorithm for Degenerate Linear Programming Problems , 1995, SIAM J. Optim..

[47]  Joaquim Júdice,et al.  An Investigation of Interior-Point Algorithms for the Linear Transportation Problem , 1996, SIAM J. Sci. Comput..

[48]  Shinji Mizuno,et al.  Convergence of a Class of Inexact Interior-Point Algorithms for Linear Programs , 1999, Math. Oper. Res..

[49]  Shinji Mizuno,et al.  Global and polynomial-time convergence of an infeasible-interior-point algorithm using inexact computation , 1997, Math. Program..