# American Institute of Mathematical Sciences

October  2010, 6(4): 825-846. doi: 10.3934/jimo.2010.6.825

## An exterior point linear programming method based on inclusive normal cones

 1 School of Mathematical & Geospatial Sciences, RMIT University, Melbourne, Australia

Received  December 2009 Revised  June 2010 Published  September 2010

In this paper, we present a geometrical exterior climbing method based on inclusive normal cones for solving general linear programming problems in canonical form. The method iteratively updates the inclusive cone by climbing within its associated inclusive region (also called a ladder), and eventually reaches an optimal solution. This method allows the development of a class of 'ladder algorithms' by using different climbing schemes. Some aspects of the current method are intrinsically related to the dual simplex method. However, it originates from different ideas and provides a new angle to look at the linear programming problem. It can be shown that the dual simplex algorithms are special ladder algorithms in this context. We present two climbing schemes leading to two finitely convergent ladder algorithms. The algorithms are tested by solving a number of linear programming examples. Some numerical results are provided.
Citation: Yanqun Liu. An exterior point linear programming method based on inclusive normal cones. Journal of Industrial & Management Optimization, 2010, 6 (4) : 825-846. doi: 10.3934/jimo.2010.6.825
##### References:
 [1] J. M. Borwein and A. S. Lewis, "Convex Analysis and Nonlinear Optimization. Theory and Examples,", Second edition. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, (2006).   Google Scholar [2] G. B. Dantzig, Maximization of a linear function of variables subject to linear inequalities,, Activity Analysis of Production and Allocation, (1951), 339.   Google Scholar [3] N. Karmarkar, A new polynomial-time algorithm for linear programming,, Combinatorica, 4 (1984), 373.  doi: 10.1007/BF02579150.  Google Scholar [4] L. G. Khachiyan, A polynomial algorithm in linear programming,, Doklady Akademii Nauk SSSR-S, 244 (1979), 1093.   Google Scholar [5] T. Terlaky and S. Zhang, Pivot rules for linear programming: A survey on recent theoretical developments,, Annals of Operations Research, 46 (1993), 203.  doi: 10.1007/BF02096264.  Google Scholar [6] R. J. Vanderbei, "Linear Programming - Foundations and Extensions,", 3rd Ed., (2008).   Google Scholar

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##### References:
 [1] J. M. Borwein and A. S. Lewis, "Convex Analysis and Nonlinear Optimization. Theory and Examples,", Second edition. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, (2006).   Google Scholar [2] G. B. Dantzig, Maximization of a linear function of variables subject to linear inequalities,, Activity Analysis of Production and Allocation, (1951), 339.   Google Scholar [3] N. Karmarkar, A new polynomial-time algorithm for linear programming,, Combinatorica, 4 (1984), 373.  doi: 10.1007/BF02579150.  Google Scholar [4] L. G. Khachiyan, A polynomial algorithm in linear programming,, Doklady Akademii Nauk SSSR-S, 244 (1979), 1093.   Google Scholar [5] T. Terlaky and S. Zhang, Pivot rules for linear programming: A survey on recent theoretical developments,, Annals of Operations Research, 46 (1993), 203.  doi: 10.1007/BF02096264.  Google Scholar [6] R. J. Vanderbei, "Linear Programming - Foundations and Extensions,", 3rd Ed., (2008).   Google Scholar
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