On projected newton barrier methods for linear programming and an equivalence to Karmarkar’s projective method
 Philip E. Gill,
 Walter Murray,
 Michael A. Saunders,
 J. A. Tomlin,
 Margaret H. Wright
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Abstract
Interest in linear programming has been intensified recently by Karmarkar’s publication in 1984 of an algorithm that is claimed to be much faster than the simplex method for practical problems. We review classical barrierfunction methods for nonlinear programming based on applying a logarithmic transformation to inequality constraints. For the special case of linear programming, the transformed problem can be solved by a “projected Newton barrier” method. This method is shown to be equivalent to Karmarkar’s projective method for a particular choice of the barrier parameter. We then present details of a specific barrier algorithm and its practical implementation. Numerical results are given for several nontrivial test problems, and the implications for future developments in linear programming are discussed.
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 Title
 On projected newton barrier methods for linear programming and an equivalence to Karmarkar’s projective method
 Journal

Mathematical Programming
Volume 36, Issue 2 , pp 183209
 Cover Date
 19860601
 DOI
 10.1007/BF02592025
 Print ISSN
 00255610
 Online ISSN
 14364646
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Linear programming
 Karmarkar’s method
 barrier methods
 Industry Sectors
 Authors

 Philip E. Gill ^{(1)}
 Walter Murray ^{(1)}
 Michael A. Saunders ^{(1)}
 J. A. Tomlin ^{(2)}
 Margaret H. Wright ^{(3)}
 Author Affiliations

 1. Department of Operations Research, Stanford University, 94305, Stanford, CA, USA
 2. Ketron Incorporated, 94040, Mountain View, California, USA
 3. Department of Operations Research, Stanford University, 94305, Stanford, CA, USA