# American Institute of Mathematical Sciences

2011, 1(3): 399-405. doi: 10.3934/naco.2011.1.399

## An improved targeted climbing algorithm for linear programs

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

Received  May 2011 Revised  July 2011 Published  September 2011

A class of exterior climbing algorithms (ladder algorithms) has been developed recently. Among this class, the targeted climbing algorithm has demonstrated very good numerical performance. In this paper an improved targeted climbing linear programming algorithm is proposed. A sequence of changing reference points, instead of a fixed reference point as in the original algorithm, is used in the improved algorithm to speed up the convergence. The improved algorithm is especially suited to solve problems involving many constraints close to the optimal solution---the case that causes many algorithms to slow down dramatically. Numerical tests and comparison with the simplex methods are presented. Our results show that the current algorithm works better on average than the simplex methods and is much faster for a large class of problems.
Citation: Mingfang Ding, Yanqun Liu, John Anthony Gear. An improved targeted climbing algorithm for linear programs. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 399-405. doi: 10.3934/naco.2011.1.399
##### References:
 [1] H. Arsham, A hybrid gradient and feasible direction pivotal solution algorithm for general linear programs, Applied Mathematics and Computation, 188 (2007), 596-611. [2] H. Arsham, T. Damij and J. Grad, An algorithm for simplex tableau reduction: the push-to-pull solution strategy, Applied Mathematics and Computation, 137 (2003), 525-547. [3] E. Barnes, V. Chen, B. Gopalakrishnan and E. L. Johnson, A least-squares primal-dual algorithm for solving linear programming problems, Operations Research Letters, 30 (2002), 289-294. doi: 10.1016/S0167-6377(02)00163-3. [4] G. B. Dantzig, "Linear Programming and Extensions," Princeton University Press, Princeton, N. J., 1963. [5] N. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica, 4 (1984), 373-395. doi: 10.1007/BF02579150. [6] Y. Liu, An exterior point linear programming method based on inclusive normal cones, Journal of Industrial and Management Optimization, 6 (2010), 825-846. doi: 10.3934/jimo.2010.6.825. [7] P. Q. Pan, A largest-distance pivot rule for the simplex algorithm, European Journal of Operational Research, 187 (2008), 393-402. [8] X. J. Xu and Y. Y. Ye, A generalized homogeneous and self-dual algorithm for linear programming, Operations Research Letters, 17 (1995), 181-190. doi: 10.1016/0167-6377(95)00002-2. [9] W. C. Yeh and H. W. Corley, A simple direct cosine simplex algorithm, Applied Mathematics and Computation, 214 (2009), 178-186. doi: 10.1016/j.amc.2009.03.080.

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##### References:
 [1] H. Arsham, A hybrid gradient and feasible direction pivotal solution algorithm for general linear programs, Applied Mathematics and Computation, 188 (2007), 596-611. [2] H. Arsham, T. Damij and J. Grad, An algorithm for simplex tableau reduction: the push-to-pull solution strategy, Applied Mathematics and Computation, 137 (2003), 525-547. [3] E. Barnes, V. Chen, B. Gopalakrishnan and E. L. Johnson, A least-squares primal-dual algorithm for solving linear programming problems, Operations Research Letters, 30 (2002), 289-294. doi: 10.1016/S0167-6377(02)00163-3. [4] G. B. Dantzig, "Linear Programming and Extensions," Princeton University Press, Princeton, N. J., 1963. [5] N. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica, 4 (1984), 373-395. doi: 10.1007/BF02579150. [6] Y. Liu, An exterior point linear programming method based on inclusive normal cones, Journal of Industrial and Management Optimization, 6 (2010), 825-846. doi: 10.3934/jimo.2010.6.825. [7] P. Q. Pan, A largest-distance pivot rule for the simplex algorithm, European Journal of Operational Research, 187 (2008), 393-402. [8] X. J. Xu and Y. Y. Ye, A generalized homogeneous and self-dual algorithm for linear programming, Operations Research Letters, 17 (1995), 181-190. doi: 10.1016/0167-6377(95)00002-2. [9] W. C. Yeh and H. W. Corley, A simple direct cosine simplex algorithm, Applied Mathematics and Computation, 214 (2009), 178-186. doi: 10.1016/j.amc.2009.03.080.
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