Exploiting special structure in Karmarkar's linear programming algorithm

We propose methods to take advantage of specially-structured constraints in a variant of Karmarkar's projective algorithm for standard form linear programming problems. We can use these constraints to generate improved bounds on the optimal value of the problem and also to compute the necessary projections more efficiently, while maintaining the theoretical bound on the algorithm's performance. It is shown how various upper-bounding constraints can be handled implicitly in this way. Unfortunately, the situation for network constraints appears less favorable.

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

[2]  G. Dantzig,et al.  The decomposition algorithm for linear programming: notes on linear programming and extensions-part 57. , 1961 .

[3]  George B. Dantzig,et al.  Linear programming and extensions , 1965 .

[4]  George B. Dantzig,et al.  Generalized Upper Bounding Techniques , 1967, J. Comput. Syst. Sci..

[5]  Leon S. Lasdon,et al.  Optimization Theory of Large Systems , 1970 .

[6]  L. Schrage.  Implicit representation of variable upper bounds in linear programming , 1975 .

[7]  Richard D. McBride,et al.  The factorization approach to large-scale linear programming , 1976, Math. Program..

[8]  Linus Schrage,et al.  Implicit representation of generalized variable upper bounds in linear programming , 1978, Math. Program..

[9]  G. Dantzig,et al.  Large-scale linear programming : proceedings of a IIASA workshop, 2-6 June 1980 , 1981 .

[10]  Bruce A. Murtagh,et al.  Advanced linear programming: Computation and practice , 1981 .

[11]  Michael J. Todd,et al.  An implementation of the simplex method for linear programming problems with variable upper bounds , 1982, Math. Program..

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

[13]  Michael A. Saunders,et al.  On projected newton barrier methods for linear programming and an equivalence to Karmarkar’s projective method , 1986, Math. Program..

[14]  Earl R. Barnes,et al.  A variation on Karmarkar’s algorithm for solving linear programming problems , 1986, Math. Program..

[15]  Pravin M. Vaidya,et al.  Fast algorithms for convex quadratic programming and multicommodity flows , 1986, STOC '86.

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

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

[18]  Michael J. Todd,et al.  Improved Bounds and Containing Ellipsoids in Karmarkar's Linear Programming Algorithm , 1988, Math. Oper. Res..

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

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