A Study of the Dual Affine Scaling Continuous Trajectories for Linear Programming

In this paper, a continuous method approach is adopted to study both the entire process and the limiting behaviors of the dual affine scaling continuous trajectories for linear programming. Our approach is different from the method presented by Adler and Monteiro (Adler and Monteiro, Math. Program. 50:29–51, 1991). Many new theoretical results on the trajectories resulting from the dual affine scaling continuous method model for linear programming are obtained.

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

[2]  Robert E. Mahony,et al.  Convergence of the Iterates of Descent Methods for Analytic Cost Functions , 2005, SIAM J. Optim..

[3]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[4]  G. Stewart.  On scaled projections and pseudoinverses , 1989 .

[5]  Renato D. C. Monteiro,et al.  Convergence and Boundary Behavior of the Projective Scaling Trajectories for Linear Programming , 1991, Math. Oper. Res..

[6]  Yvonne Freeh,et al.  Interior Point Algorithms Theory And Analysis , 2016 .

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

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

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

[10]  Michael J. Todd,et al.  A Dantzig-Wolfe-Like Variant of Karmarkar's Interior-Point Linear Programming Algorithm , 1990, Oper. Res..

[11]  Jean-Philippe Vial,et al.  Theory and algorithms for linear optimization - an interior point approach , 1998, Wiley-Interscience series in discrete mathematics and optimization.

[12]  Liqun Qi,et al.  Neurodynamical Optimization , 2004, J. Glob. Optim..

[13]  Clóvis C. Gonzaga,et al.  Interior point algorithms for linear programming with inequality constraints , 1991, Math. Program..

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

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

[16]  W. Rudin.  Principles of mathematical analysis , 1964 .

[17]  Moody T. Chu,et al.  Dynamical System Characterization of the Central Path and Its Variants - A Revisit , 2011, SIAM J. Appl. Dyn. Syst..

[18]  T Talaky,et al.  Interior Point Methods of Mathematical Programming , 1997 .

[19]  Gene H. Golub,et al.  Continuous methods for extreme and interior eigenvalue problems , 2006 .

[20]  Takashi Tsuchiya,et al.  Affine Scaling Algorithm , 1996 .

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

[22]  Weiping Li,et al.  Applied Nonlinear Control , 1991 .