A Weaker Constraint Qualification of Globally Convergent Homotopy Method for a Multiobjective Programming Problem

Affiliation(s)

Department of Mathematics, Clarkson University, Potsdam, USA.

Department of Mathematics, Harbin Normal University, Harbin, China.

Department of Mathematics, Clarkson University, Potsdam, USA.

Department of Mathematics, Harbin Normal University, Harbin, China.

ABSTRACT

In this paper, we prove that the combined homotopy interior point method for a multiobjective programming problem introduced in Ref. [1] remains valid under a weaker constrained qualification—the Mangasarian-Fromovitz constrained qualification, instead of linear independence constraint qualification. The algorithm generated by this method associated to the Karush-Kuhn-Tucker points of the multiobjective programming problem is proved to be globally convergent.

KEYWORDS

Multiobjective Programming Problem; Homotopy Method; KKT Condition; Efficient Solution; MFCQ

Multiobjective Programming Problem; Homotopy Method; KKT Condition; Efficient Solution; MFCQ

Cite this paper

G. Yao and W. Song, "A Weaker Constraint Qualification of Globally Convergent Homotopy Method for a Multiobjective Programming Problem,"*Applied Mathematics*, Vol. 4 No. 2, 2013, pp. 343-347. doi: 10.4236/am.2013.42052.

G. Yao and W. Song, "A Weaker Constraint Qualification of Globally Convergent Homotopy Method for a Multiobjective Programming Problem,"

References

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

[2] 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.

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

[4] C. Y. Lin and J. L. Dong, “Methods and Theories in Multiobjective Optimization,” Jinlin Education Press, Chang chun, 1992.

[5] M. Abadie, “Generalized Kuhn Tucker Conditions for Mathematical Programming,” SIAM Journal on Control, Vol. 7, No. 2, 1969, pp. 232 241. doi:10.1137/0307016

[6] R. B. Kellogg, T. Y. Li and J. A. Yorke, “A Constructive Proof of the Brouwer Fixed Point Theorem and Computational Results,” SIAM Journal on Numerical Analysis, Vol. 13, No. 4, 1976, pp. 473 483. doi:10.1137/0713041

[7] S. N. Chow, J. Mallet Paret and J. A. Yorke, “Finding Zeros of Maps: Homotopy Methods That are Constructive with Probability One,” Mathematical Computation, Vol. 32, 1978, pp. 887 899. doi:10.1090/S0025 5718 1978 0492046 9

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

[9] 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.

[10] E. L. Allgower and K. Georg, “Numerical Continuation Methods: An Introduction,” Springer Verlag, Berlin, 1990. doi:10.1007/978 3 642 61257 2

[11] Z. H. Lin, B. Yu and G. C. Feng, “A Combined Homotopy Interior Method for Convex Nonlinear Programming,” Applied Mathematics and Computation, Vol. 84, No. 2 3, 1997, pp. 193 211. doi:10.1016/S0096 3003(96)00086 0

[12] 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, 1996, pp. 209 224. doi:10.1016/0096 3003(95)00295 2

[13] Z. H. Lin, D. L. Zhu and Z. P. Sheng, “Finding a Minimal Efficient Solution of a Convex Multiobjective Program,” Journal of Optimization Theory and Applications, Vol. 118, No. 1, 2003, pp. 587 600. doi:10.1023/A:1024739508603

[14] Y. F. Shang and B. Yu, “A Constraint Shifting Homotopy Method for Convex Multi Objective Programming,” Journal of Computational and Applied Mathematics, Vol. 236, No. 5, 2011, pp. 640 646.

[15] X. Zhao, S. G. Zhang and Q. H. Liu, “Homotopy Interior Point Method for a General Multiobjective Programming Problem,” Journal of Applied Mathematics, Vol. 2012, 2012, Article ID: 497345.

[16] N. Kim and L. Thuy, “An Algorithm for Generating Efficient Outcome Points for Convex Multiobjective Programming Problem,” Intelligent Information and Database Systems Lecture Notes in Computer Science, Vol. 5991, No. 2010, 2010, pp. 390 399.

[17] Z. Chen, “Multiobjective Optimization Problems, Vector Variational Inequalities and Proximal Type Methods,” Dissertations, Hong Kong Polytechnic University, Kowloon, 2010.

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

[2] 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.

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

[4] C. Y. Lin and J. L. Dong, “Methods and Theories in Multiobjective Optimization,” Jinlin Education Press, Chang chun, 1992.

[5] M. Abadie, “Generalized Kuhn Tucker Conditions for Mathematical Programming,” SIAM Journal on Control, Vol. 7, No. 2, 1969, pp. 232 241. doi:10.1137/0307016

[6] R. B. Kellogg, T. Y. Li and J. A. Yorke, “A Constructive Proof of the Brouwer Fixed Point Theorem and Computational Results,” SIAM Journal on Numerical Analysis, Vol. 13, No. 4, 1976, pp. 473 483. doi:10.1137/0713041

[7] S. N. Chow, J. Mallet Paret and J. A. Yorke, “Finding Zeros of Maps: Homotopy Methods That are Constructive with Probability One,” Mathematical Computation, Vol. 32, 1978, pp. 887 899. doi:10.1090/S0025 5718 1978 0492046 9

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

[9] 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.

[10] E. L. Allgower and K. Georg, “Numerical Continuation Methods: An Introduction,” Springer Verlag, Berlin, 1990. doi:10.1007/978 3 642 61257 2

[11] Z. H. Lin, B. Yu and G. C. Feng, “A Combined Homotopy Interior Method for Convex Nonlinear Programming,” Applied Mathematics and Computation, Vol. 84, No. 2 3, 1997, pp. 193 211. doi:10.1016/S0096 3003(96)00086 0

[12] 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, 1996, pp. 209 224. doi:10.1016/0096 3003(95)00295 2

[13] Z. H. Lin, D. L. Zhu and Z. P. Sheng, “Finding a Minimal Efficient Solution of a Convex Multiobjective Program,” Journal of Optimization Theory and Applications, Vol. 118, No. 1, 2003, pp. 587 600. doi:10.1023/A:1024739508603

[14] Y. F. Shang and B. Yu, “A Constraint Shifting Homotopy Method for Convex Multi Objective Programming,” Journal of Computational and Applied Mathematics, Vol. 236, No. 5, 2011, pp. 640 646.

[15] X. Zhao, S. G. Zhang and Q. H. Liu, “Homotopy Interior Point Method for a General Multiobjective Programming Problem,” Journal of Applied Mathematics, Vol. 2012, 2012, Article ID: 497345.

[16] N. Kim and L. Thuy, “An Algorithm for Generating Efficient Outcome Points for Convex Multiobjective Programming Problem,” Intelligent Information and Database Systems Lecture Notes in Computer Science, Vol. 5991, No. 2010, 2010, pp. 390 399.

[17] Z. Chen, “Multiobjective Optimization Problems, Vector Variational Inequalities and Proximal Type Methods,” Dissertations, Hong Kong Polytechnic University, Kowloon, 2010.