American Institute of Mathematical Sciences

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April  2014, 10(2): 397-412. doi: 10.3934/jimo.2014.10.397

A ladder method for linear semi-infinite programming

Yanqun Liu 1, and  Ming-Fang Ding 1,

1. 

School of Mathematical and Geospatial Sciences, RMIT University, GPO Box 2476, Melbourne, Victoria 3001, Australia, Australia

Received  November 2012 Revised  July 2013 Published  October 2013

This paper presents a new method for linear semi-infinite programming. With the introduction of the so-called generalized ladder point, a ladder method for linear semi-infinite programming is developed. This work includes the generalization of the inclusive cone version of the fundamental theorem of linear programming and the extension of a linear programming ladder algorithm. The extended ladder algorithm finds a generalized ladder point optimal solution of the linear semi-infinite programming problem, which is approximated by a sequence of ladder points. Simple convergence properties are provided. The algorithm is tested by solving a number of linear semi-infinite programming examples. These numerical results indicate that the algorithm is very efficient when compared with other methods.
Keywords: optimality, ladder, inclusive cone, Linear semi-infinite programming, algorithm..
Mathematics Subject Classification: Primary: 90C34, 90C4.
Citation: Yanqun Liu, Ming-Fang Ding. A ladder method for linear semi-infinite programming. Journal of Industrial & Management Optimization, 2014, 10 (2) : 397-412. doi: 10.3934/jimo.2014.10.397
References:
[1]

E. J. Anderson and A. S. Lewis, An extension of the simplex algorithm for semi-infinite linear programming, Mathematical Programming, 44 (1989), 247-269. doi: 10.1007/BF01587092.  Google Scholar

[2]

B. Betrò, An accelerated central cutting plane algorithm for semi-infinite linear programming, Mathematical Programming, 101 (2004), 479-495. doi: 10.1007/s10107-003-0492-5.  Google Scholar

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D. den Hertog, J. Kaliski, C. Roos and T. Terlaky, A logarithmic barrier cutting plane method for convex programming, Annals of Operations Research, 58 (1995), 69-98. doi: 10.1007/BF02032162.  Google Scholar

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M. C. Ferris and A. B. Philpott, An interior point algorithm for semi-infinite linear programming, Mathematical Programming, 43 (1989), 257-276. doi: 10.1007/BF01582293.  Google Scholar

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M. A. Goberna and M. A. López, Linear Semi-infinite Optimization, Wiley Series in Mathematical Methods in Practice, 2, John Wiley & Sons, Ltd., Chichester, 1998.  Google Scholar

[6]

M. A. Goberna, Linear semi-infinite optimization: Recent advances, in Continuous Optimization (eds. V. Jeyakumar and A. M. Rubinov), Appl. Optim., 99, Springer, New York, 2005, 3-22. doi: 10.1007/0-387-26771-9_1.  Google Scholar

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R. Hettich, A review of numerical methods for semi-infinite optimization, in Semi-infinite Programming and Applications (eds. A. V. Fiacco and K. O. Kortanek), Lecture Notes in Econom. and Math. Systems, 215, Springer, Berlin, 1983, 158-178. doi: 10.1007/978-3-642-46477-5_11.  Google Scholar

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R. Hettich, An implementation of a discretization method for semi-infinite programming, Mathematical Programming, 34 (1986), 354-361. doi: 10.1007/BF01582235.  Google Scholar

[9]

R. Hettich and K. O. Kortanek, Semi-infinite programming: Theory, methods, and applications, SIAM Rev., 35 (1993), 380-429. doi: 10.1137/1035089.  Google Scholar

[10]

S. Ito, Y. Liu and K. L. Teo, A dual parametrization method for convex semi-infinite programming, Ann. Oper. Res., 98 (2000), 189-213. doi: 10.1023/A:1019208524259.  Google Scholar

[11]

A. Kaplan and R. Tichatschke, Proximal interior point method for convex semi-infinite programming, Optim. Methods Softw., 15 (2001), 87-119. doi: 10.1080/10556780108805813.  Google Scholar

[12]

Y. Liu, An exterior point method for linear programming based on inclusive normal cones, J. Ind. Manag. Optim., 6 (2010), 825-846. doi: 10.3934/jimo.2010.6.825.  Google Scholar

[13]

Y. Liu, Duality theorem in linear programming: From trichotomy to quadrichotomy, J. Ind. Manag. Optim., 7 (2011), 1003-1011. doi: 10.3934/jimo.2011.7.1003.  Google Scholar

[14]

Y. Liu and K. L. Teo, A bridging method for global optimization, J. Austral. Math. Soc. Ser. B, 41 (1999), 41-57. doi: 10.1017/S0334270000011024.  Google Scholar

[15]

Y. Liu, K. L. Teo and S. Y. Wu, A New quadratic semi-infinite programming algorithm based on dual parametrization, J. Global Optim., 29 (2004), 401-413. doi: 10.1023/B:JOGO.0000047910.80739.95.  Google Scholar

[16]

R. Reemtsen, Discretization methods for the solution of semi-infinite programming problems, J. Optim. Theory Appl., 71 (1991), 85-103. doi: 10.1007/BF00940041.  Google Scholar

[17]

G. A. Watson, Lagrangian methods for semi-infinite programming problems, in Infinite Programming, Lecture Notes in Economics and Mathematical Systems (eds. E. J. Anderson and A. B. Philpott), Lecture Notes in Econom. and Math. Systems, 259, Springer, Berlin, 1985, 90-107. doi: 10.1007/978-3-642-46564-2_8.  Google Scholar

[18]

S. Y. Wu, S. C. Fang and C. J. Lin, Relaxed cutting plane method for solving linear semi-infinite programming problems, J. Optim. Theory Appl., 99 (1998), 759-779. doi: 10.1023/A:1021763419562.  Google Scholar

show all references

References:
[1]

E. J. Anderson and A. S. Lewis, An extension of the simplex algorithm for semi-infinite linear programming, Mathematical Programming, 44 (1989), 247-269. doi: 10.1007/BF01587092.  Google Scholar

[2]

B. Betrò, An accelerated central cutting plane algorithm for semi-infinite linear programming, Mathematical Programming, 101 (2004), 479-495. doi: 10.1007/s10107-003-0492-5.  Google Scholar

[3]

D. den Hertog, J. Kaliski, C. Roos and T. Terlaky, A logarithmic barrier cutting plane method for convex programming, Annals of Operations Research, 58 (1995), 69-98. doi: 10.1007/BF02032162.  Google Scholar

[4]

M. C. Ferris and A. B. Philpott, An interior point algorithm for semi-infinite linear programming, Mathematical Programming, 43 (1989), 257-276. doi: 10.1007/BF01582293.  Google Scholar

[5]

M. A. Goberna and M. A. López, Linear Semi-infinite Optimization, Wiley Series in Mathematical Methods in Practice, 2, John Wiley & Sons, Ltd., Chichester, 1998.  Google Scholar

[6]

M. A. Goberna, Linear semi-infinite optimization: Recent advances, in Continuous Optimization (eds. V. Jeyakumar and A. M. Rubinov), Appl. Optim., 99, Springer, New York, 2005, 3-22. doi: 10.1007/0-387-26771-9_1.  Google Scholar

[7]

R. Hettich, A review of numerical methods for semi-infinite optimization, in Semi-infinite Programming and Applications (eds. A. V. Fiacco and K. O. Kortanek), Lecture Notes in Econom. and Math. Systems, 215, Springer, Berlin, 1983, 158-178. doi: 10.1007/978-3-642-46477-5_11.  Google Scholar

[8]

R. Hettich, An implementation of a discretization method for semi-infinite programming, Mathematical Programming, 34 (1986), 354-361. doi: 10.1007/BF01582235.  Google Scholar

[9]

R. Hettich and K. O. Kortanek, Semi-infinite programming: Theory, methods, and applications, SIAM Rev., 35 (1993), 380-429. doi: 10.1137/1035089.  Google Scholar

[10]

S. Ito, Y. Liu and K. L. Teo, A dual parametrization method for convex semi-infinite programming, Ann. Oper. Res., 98 (2000), 189-213. doi: 10.1023/A:1019208524259.  Google Scholar

[11]

A. Kaplan and R. Tichatschke, Proximal interior point method for convex semi-infinite programming, Optim. Methods Softw., 15 (2001), 87-119. doi: 10.1080/10556780108805813.  Google Scholar

[12]

Y. Liu, An exterior point method for linear programming based on inclusive normal cones, J. Ind. Manag. Optim., 6 (2010), 825-846. doi: 10.3934/jimo.2010.6.825.  Google Scholar

[13]

Y. Liu, Duality theorem in linear programming: From trichotomy to quadrichotomy, J. Ind. Manag. Optim., 7 (2011), 1003-1011. doi: 10.3934/jimo.2011.7.1003.  Google Scholar

[14]

Y. Liu and K. L. Teo, A bridging method for global optimization, J. Austral. Math. Soc. Ser. B, 41 (1999), 41-57. doi: 10.1017/S0334270000011024.  Google Scholar

[15]

Y. Liu, K. L. Teo and S. Y. Wu, A New quadratic semi-infinite programming algorithm based on dual parametrization, J. Global Optim., 29 (2004), 401-413. doi: 10.1023/B:JOGO.0000047910.80739.95.  Google Scholar

[16]

R. Reemtsen, Discretization methods for the solution of semi-infinite programming problems, J. Optim. Theory Appl., 71 (1991), 85-103. doi: 10.1007/BF00940041.  Google Scholar

[17]

G. A. Watson, Lagrangian methods for semi-infinite programming problems, in Infinite Programming, Lecture Notes in Economics and Mathematical Systems (eds. E. J. Anderson and A. B. Philpott), Lecture Notes in Econom. and Math. Systems, 259, Springer, Berlin, 1985, 90-107. doi: 10.1007/978-3-642-46564-2_8.  Google Scholar

[18]

S. Y. Wu, S. C. Fang and C. J. Lin, Relaxed cutting plane method for solving linear semi-infinite programming problems, J. Optim. Theory Appl., 99 (1998), 759-779. doi: 10.1023/A:1021763419562.  Google Scholar

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