Solving combinatorial optimization problems using Karmarkar's algorithm

We describe a cutting plane algorithm for solving combinatorial optimization problems. The primal projective standard-form variant of Karmarkar's algorithm for linear programming is applied to the duals of a sequence of linear programming relaxations of the combinatorial optimization problem.

[1]  J. Goffin.  Affine and projective transformations in nondifferentiable optimization , 1988 .

[2]  Clóvis C. Gonzaga,et al.  Conical projection algorithms for linear programming , 1989, Math. Program..

[3]  Gene H. Golub,et al.  Matrix computations , 1983 .

[4]  J. Goffin,et al.  Decomposition and nondifferentiable optimization with the projective algorithm , 1992 .

[5]  Mauricio G. C. Resende,et al.  An implementation of Karmarkar's algorithm for linear programming , 1989, Math. Program..

[6]  T. C. Hu,et al.  Multi-Terminal Network Flows , 1961 .

[7]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, STOC '84.

[8]  Robert M. Freund,et al.  ALGORITHM FOR SOLVING A LINEAR PROGRAM DIRECTLY FROM AN INFEASIBLE "WARM START" , 1991 .

[9]  Nicos Christofides.  Worst-Case Analysis of a New Heuristic for the Travelling Salesman Problem , 1976, Operations Research Forum.

[10]  David M. Gay,et al.  A variant of Karmarkar's linear programming algorithm for problems in standard form , 1987, Math. Program..

[11]  A. George,et al.  Orthogonal Reduction of Sparse Matrices to Upper Triangular Form Using Householder Transformations , 1986 .

[12]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..

[13]  M. R. Rao,et al.  Odd Minimum Cut-Sets and b-Matchings , 1982, Math. Oper. Res..

[14]  Yinyu Ye,et al.  A Potential Reduction Algorithm Allowing Column Generation , 1992, SIAM J. Optim..

[15]  D. Bayer,et al.  The nonlinear geometry of linear programming. II. Legendre transform coordinates and central trajectories , 1989 .

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

[17]  Clyde L. Monma,et al.  Further Development of a Primal-Dual Interior Point Method , 1990, INFORMS J. Comput..

[18]  J. Goffin,et al.  Cutting planes and column generation techniques with the projective algorithm , 1990 .

[19]  Nimrod Megiddo,et al.  Boundary Behavior of Interior Point Algorithms in Linear Programming , 1989, Math. Oper. Res..

[20]  Narendra Karmarkar,et al.  An Interior-Point Approach to NP-Complete Problems , 1990, Conference on Integer Programming and Combinatorial Optimization.

[21]  J. Mitchell.  Karmarkar's algorithm and combinatorial optimization problems , 1988 .

[22]  Kees Roos,et al.  A survey of search directions in interior point methods for linear programming , 1991, Math. Program..

[23]  John E. Mitchell.  An Interior Point Column Generation Method for Linear Programming Using Shifted Barriers , 1994, SIAM J. Optim..

[24]  M. Todd,et al.  Chapter II Linear programming , 1989 .

[25]  Shen Lin.  Computer solutions of the traveling salesman problem , 1965 .

[26]  Roy E. Marsten,et al.  On Implementing Mehrotra's Predictor-Corrector Interior-Point Method for Linear Programming , 1992, SIAM J. Optim..

[27]  M. Grötschel,et al.  Solving matching problems with linear programming , 1985, Math. Program..

[28]  Angelika Steger,et al.  An Extension of Karmarkar’s Algorithm for Bounded Linear Programming Problems , 1988 .

[29]  Jack Edmonds,et al.  Maximum matching and a polyhedron with 0,1-vertices , 1965 .

[30]  I. Lustig,et al.  Computational experience with a primal-dual interior point method for linear programming , 1991 .

[31]  Mauricio G. C. Resende,et al.  An interior point algorithm to solve computationally difficult set covering problems , 1991, Math. Program..

[32]  D. Bayer,et al.  The Non-Linear Geometry of Linear Pro-gramming I: A?ne and projective scaling trajectories , 1989 .

[33]  J. Edmonds.  Paths, Trees, and Flowers , 1965, Canadian Journal of Mathematics.

[34]  A. George,et al.  Solution of sparse linear least squares problems using givens rotations , 1980 .

[35]  Masakazu Kojima,et al.  Recovering optimal dual solutions in Karmarkar's polynomial algorithm for linear programming , 1987, Math. Program..

[36]  Michael J. Todd,et al.  A variant of Karmarkar's linear programming algorithm for problems with some unrestricted variables , 1989 .

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