Homotopy Continuous Method for Weak Efficient Solution of Multiobjective Optimization Problem with Feasible Set Unbounded Condition

ABSTRACT

In this paper, we propose a homotopy continuous method (HCM) for solving a weak efficient solution of multiobjective optimization problem (MOP) with feasible set unbounded condition, which is arising in Economical Distributions, Engineering Decisions, Resource Allocations and other field of mathematical economics and engineering problems. Under the suitable assumption, it is proved to globally converge to a weak efficient solution of (MOP), if its x-branch has no weak infinite solution.

In this paper, we propose a homotopy continuous method (HCM) for solving a weak efficient solution of multiobjective optimization problem (MOP) with feasible set unbounded condition, which is arising in Economical Distributions, Engineering Decisions, Resource Allocations and other field of mathematical economics and engineering problems. Under the suitable assumption, it is proved to globally converge to a weak efficient solution of (MOP), if its x-branch has no weak infinite solution.

Cite this paper

W. Xing and B. Wu, "Homotopy Continuous Method for Weak Efficient Solution of Multiobjective Optimization Problem with Feasible Set Unbounded Condition,"*Applied Mathematics*, Vol. 3 No. 7, 2012, pp. 765-771. doi: 10.4236/am.2012.37114.

W. Xing and B. Wu, "Homotopy Continuous Method for Weak Efficient Solution of Multiobjective Optimization Problem with Feasible Set Unbounded Condition,"

References

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[2] G. Fandel and T. Gal, “Multiple Criteria Decision Making: Theory and Applications,” Springer-Verlag Press, New York, 1980. doi:10.1007/978-3-642-48782-8

[3] C. B. Garcia and W. I. Zangwill, “Pathways to Solutions, Fixed Points and Equilibria,” Prentice-Hall, Englewood Cliffs, 1981, pp. 475-495.

[4] N. Megiddo, “Pathways to the Optimal Set in Linear Programming,” In: N. Megiddo, Ed., Progress in Mathematical Programming Interior Point and Related Methods, Springer, New York, 1988, pp. 131-158.

[5] M. Kojima, S. Mizuno and A. Yoshise, “A Primal-Dual Interior Point Algorithm for Linear Programming,” In: N. Megiddo, Ed., Progress in Mathematical Programming Interior Point and Related Methods, Springer, New York, 1988, pp. 29-47.

[6] Z. H. Lin, B. Yu and G. C. Feng, “A Combined Homotopy Interior Point Method for Convex Nonlinear Programming Problems,” Applied Mathematics and Computation, Vol. 84, No. 2-3, 1994, pp. 193-211. doi:10.1016/S0096-3003(96)00086-0

[7] Z. H. Lin, Y. Li and B. Yu, “A Combined Homotopy Interior Point Method for General Nonlinear Programming Problems,” Applied Mathematics and Computation, Vol. 80, No. 2-3, 1997, pp. 209-224. doi:10.1016/0096-3003(95)00295-2

[8] Z. H. Lin, D. L. Zhu and Z. D. Sheng, “Finding a Minimal Efficient Solution of a Convex Multiobjective Program,” Journal of Optimization Theory and Applications, Vol. 118, No. 3, 2003, pp. 587-600. doi:10.1023/B:JOTA.0000004872.93803.09

[9] W. Song and G. M. Yao, “Homotopy Method for a General Multiobjective Programming Problem,” Journal of Optimization Theory and Applications, Vol. 138, No. 1, 2008, pp. 139-153. doi:10.1007/s10957-008-9366-6

[10] T. Maeda, “Second-Order Conditions for Efficiency Nonsmooth Multiobjective Optimization Problems,” Journal of Optimization Theory and Applications, Vol. 122, No. 3, 2004, pp. 521-538. doi:10.1023/B:JOTA.0000042594.46637.b4

[11] L. T. Watson, “Globally Convergent Homotopy Algorithms for Nonlinear Systems of Equations,” Nonlinear Dynamics, Vol. 1, No. 2, 1990, pp.143-191. doi:10.1007/BF01857785

[12] G. C. Feng, Z. L. Lin and B. Yu, “Existence of an Interior Pathway to a Karush-Kuhn-Tucker Point of a Nonlinear Programming Problem,” Nonlinear Analysis, Vol. 32, No. 6, 1998, pp. 761-768. doi:10.1016/S0362-546X(97)00516-6

[13] Q. Xu, B. Yu and G. C. Feng, “Homotopy Methods for Solving Variational Inequalities in Unbounded Sets,” Journal of Global Optimization, Vol. 31, No. 1, 2005, pp. 121-131. doi:10.1007/s10898-004-4272-4

[14] Q. Xu, B. Yu, G. C. Feng and C. Y. Dan, “Condition for Global Convergence of a Homotopy Method for Variational Inequality Problems on Unbounded Sets,” Optimization Methods and Software, Vol. 22, No. 4, 2007, pp. 587-599. doi:10.1080/10556780600887883

[15] C. Y. Lin and J. L. Dong, “Methods and Theories in Multiobjective Optimization,” Jilin Education Press, Changchun, 1992.

[16] H. W. Kuhn and A. W. Tucker, “Nonlinear Programming,” Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, 1951.

[17] G. L. Naber, “Topological Methods in Euclidean Space,” Cambridge University Press, London, 1980.

[1] A. Charne and W. W. Cooper, “Management Models and Industrial Application of Linear Programming,” Wiley Press, New York, 1961.

[2] G. Fandel and T. Gal, “Multiple Criteria Decision Making: Theory and Applications,” Springer-Verlag Press, New York, 1980. doi:10.1007/978-3-642-48782-8

[3] C. B. Garcia and W. I. Zangwill, “Pathways to Solutions, Fixed Points and Equilibria,” Prentice-Hall, Englewood Cliffs, 1981, pp. 475-495.

[4] N. Megiddo, “Pathways to the Optimal Set in Linear Programming,” In: N. Megiddo, Ed., Progress in Mathematical Programming Interior Point and Related Methods, Springer, New York, 1988, pp. 131-158.

[5] M. Kojima, S. Mizuno and A. Yoshise, “A Primal-Dual Interior Point Algorithm for Linear Programming,” In: N. Megiddo, Ed., Progress in Mathematical Programming Interior Point and Related Methods, Springer, New York, 1988, pp. 29-47.

[6] Z. H. Lin, B. Yu and G. C. Feng, “A Combined Homotopy Interior Point Method for Convex Nonlinear Programming Problems,” Applied Mathematics and Computation, Vol. 84, No. 2-3, 1994, pp. 193-211. doi:10.1016/S0096-3003(96)00086-0

[7] Z. H. Lin, Y. Li and B. Yu, “A Combined Homotopy Interior Point Method for General Nonlinear Programming Problems,” Applied Mathematics and Computation, Vol. 80, No. 2-3, 1997, pp. 209-224. doi:10.1016/0096-3003(95)00295-2

[8] Z. H. Lin, D. L. Zhu and Z. D. Sheng, “Finding a Minimal Efficient Solution of a Convex Multiobjective Program,” Journal of Optimization Theory and Applications, Vol. 118, No. 3, 2003, pp. 587-600. doi:10.1023/B:JOTA.0000004872.93803.09

[9] W. Song and G. M. Yao, “Homotopy Method for a General Multiobjective Programming Problem,” Journal of Optimization Theory and Applications, Vol. 138, No. 1, 2008, pp. 139-153. doi:10.1007/s10957-008-9366-6

[10] T. Maeda, “Second-Order Conditions for Efficiency Nonsmooth Multiobjective Optimization Problems,” Journal of Optimization Theory and Applications, Vol. 122, No. 3, 2004, pp. 521-538. doi:10.1023/B:JOTA.0000042594.46637.b4

[11] L. T. Watson, “Globally Convergent Homotopy Algorithms for Nonlinear Systems of Equations,” Nonlinear Dynamics, Vol. 1, No. 2, 1990, pp.143-191. doi:10.1007/BF01857785

[12] G. C. Feng, Z. L. Lin and B. Yu, “Existence of an Interior Pathway to a Karush-Kuhn-Tucker Point of a Nonlinear Programming Problem,” Nonlinear Analysis, Vol. 32, No. 6, 1998, pp. 761-768. doi:10.1016/S0362-546X(97)00516-6

[13] Q. Xu, B. Yu and G. C. Feng, “Homotopy Methods for Solving Variational Inequalities in Unbounded Sets,” Journal of Global Optimization, Vol. 31, No. 1, 2005, pp. 121-131. doi:10.1007/s10898-004-4272-4

[14] Q. Xu, B. Yu, G. C. Feng and C. Y. Dan, “Condition for Global Convergence of a Homotopy Method for Variational Inequality Problems on Unbounded Sets,” Optimization Methods and Software, Vol. 22, No. 4, 2007, pp. 587-599. doi:10.1080/10556780600887883

[15] C. Y. Lin and J. L. Dong, “Methods and Theories in Multiobjective Optimization,” Jilin Education Press, Changchun, 1992.

[16] H. W. Kuhn and A. W. Tucker, “Nonlinear Programming,” Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, 1951.

[17] G. L. Naber, “Topological Methods in Euclidean Space,” Cambridge University Press, London, 1980.