A simplified global convergence proof of the affine scaling algorithm

This paper presents a simplified and self-contained global convergence proof for the affine scaling algorithm applied to degenerate linear programming problems. Convergence of the sequence of dual estimates to the center of the optimal dual face is also proven. In addition, we give a sharp rate of convergence result for the sequence of objective function values. All these results are proved with respect to the long step version of the affine scaling algorithm in which we move a fraction λ, where λ ∈ (0,2/3), of the step to the boundary of the feasible region.

[1]  I. Dikin Determining the interior point of a system of linear inequalities , 1992 .

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

[3]  Clyde L. Monma,et al.  Computational experience with a dual affine variant of Karmarkar's method for linear programming , 1987 .

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

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

[6]  A. Hoffman On approximate solutions of systems of linear inequalities , 1952 .

[7]  R. Vanderbei Dikin ' s Convergence Result for the Affine-Scaling Algorithm , 1990 .

[8]  P. Boggs,et al.  On the convergence behavior of trajectories for linear programming , 1988 .

[9]  Takashi Tsuchiya,et al.  Global Convergence Property of the Affine Scaling Methods for Primal Degenerate Linear Programming Problems , 1992, Math. Oper. Res..

[10]  Paul Tseng,et al.  On the convergence of the affine-scaling algorithm , 1992, Math. Program..

[11]  Takashi Tsuchiya,et al.  Global convergence of the affine scaling methods for degenerate linear programming problems , 1991, Math. Program..

[12]  Robert J. Vanderbei,et al.  Two-thirds is sharp for affine scaling , 1993, Oper. Res. Lett..

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

[14]  A. I. Perov,et al.  On the convergence of an iterative process , 1977 .

[15]  M. Todd,et al.  Mathematical Developments Arising from Linear Programming , 1990 .

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

[17]  Renato D. C. Monteiro,et al.  Limiting behavior of the affine scaling continuous trajectories for linear programming problems , 1991, Math. Program..