American Institute of Mathematical Sciences

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October  2011, 7(4): 1003-1011. doi: 10.3934/jimo.2011.7.1003

Duality in linear programming: From trichotomy to quadrichotomy

Yanqun Liu 1,

1. 

School of Mathematical & Geospatial Sciences, RMIT University, Melbourne

Received  September 2010 Revised  July 2011 Published  August 2011

In this paper, we present a new approach to the duality of linear programming. We extend the boundedness to the so called inclusiveness, and show that inclusiveness and feasibility are a pair of coexisting and mutually dual properties in linear programming: one of them is possessed by a primal problem if and only if the other is possessed by the dual problem. This duality relation is consistent with the symmetry between the primal and dual problems and leads to a duality result that is considered a completion of the classical strong duality theorem. From this result, complete solvability information of the primal (or dual) problem can be derived solely from dual (or primal) information. This is demonstrated by applying the new duality results to a recent linear programming method.
Keywords: Linear programming, inclusive normal cone, inclusiveness., duality.
Mathematics Subject Classification: Primary: 90C05; Secondary: 49M2.
Citation: Yanqun Liu. Duality in linear programming: From trichotomy to quadrichotomy. Journal of Industrial & Management Optimization, 2011, 7 (4) : 1003-1011. doi: 10.3934/jimo.2011.7.1003
References:
[1]

M. C. Ferris, O. L. Mangasarian and S. J. Wright, "Linear Programming with MATLAB,", MPS-SIAM Series on Optimization, 7 (2007).  doi: 10.1137/1.9780898718775.  Google Scholar

[2]

H. J. Greenberg, How to analyze the results of linear programs--part 3: Infeasibility diagnosis,, Interface, 23 (1993), 120.  doi: 10.1287/inte.23.6.120.  Google Scholar

[3]

C. Li, X. He, B. Chen, Z. Gong, B. Chen and Q. Zhang, Infeasibility diagnosis on the linear programming model of production planning in refinery,, Chinese J. Chem. Eng., 14 (2006), 569.  doi: 10.1016/S1004-9541(06)60117-1.  Google Scholar

[4]

Y. Liu, An exterior point linear programming method based on inclusive normal cones,, Journal of Industrial and Management Optimization, 6 (2010), 825.  doi: 10.3934/jimo.2010.6.825.  Google Scholar

[5]

D. G. Luenburg and Y. Ye, "Linear and Nonlinear Programming,", 3rd edition, 116 (2008).   Google Scholar

[6]

G. Roodman, Post-infeasibility analysis in linear programming,, Management Science, 25 (1979), 916.  doi: 10.1287/mnsc.25.9.916.  Google Scholar

show all references

References:
[1]

M. C. Ferris, O. L. Mangasarian and S. J. Wright, "Linear Programming with MATLAB,", MPS-SIAM Series on Optimization, 7 (2007).  doi: 10.1137/1.9780898718775.  Google Scholar

[2]

H. J. Greenberg, How to analyze the results of linear programs--part 3: Infeasibility diagnosis,, Interface, 23 (1993), 120.  doi: 10.1287/inte.23.6.120.  Google Scholar

[3]

C. Li, X. He, B. Chen, Z. Gong, B. Chen and Q. Zhang, Infeasibility diagnosis on the linear programming model of production planning in refinery,, Chinese J. Chem. Eng., 14 (2006), 569.  doi: 10.1016/S1004-9541(06)60117-1.  Google Scholar

[4]

Y. Liu, An exterior point linear programming method based on inclusive normal cones,, Journal of Industrial and Management Optimization, 6 (2010), 825.  doi: 10.3934/jimo.2010.6.825.  Google Scholar

[5]

D. G. Luenburg and Y. Ye, "Linear and Nonlinear Programming,", 3rd edition, 116 (2008).   Google Scholar

[6]

G. Roodman, Post-infeasibility analysis in linear programming,, Management Science, 25 (1979), 916.  doi: 10.1287/mnsc.25.9.916.  Google Scholar

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