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Homotopy Method for a General Multiobjective Programming Problem

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Abstract

In this paper, we present a combined homotopy interior-point method for a general multiobjective programming problem. The algorithm generated by this method associated to Karush–Kuhn–Tucker points of the multiobjective programming problem is proved to be globally convergent under some basic assumptions.

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References

  1. Kuhn, H.W., Tucker, A.W.: Nonlinear programming. In: Neyman, J. (ed.) Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, Berkeley (1951)

    Google Scholar 

  2. Abadie, M.: Generalized Kuhn–Tucker conditions for mathematical programming. SIAM J. Control. 7, 232–241 (1969)

    Article  MathSciNet  Google Scholar 

  3. Lin, C.Y., Dong, J.L.: Methods and Theories in Multiobjective Optimization. Jinlin Education Press, Changchun (1992) (in Chinese)

    Google Scholar 

  4. Maeda, T.: Second-order conditions for efficiency in nonsmmoth multiobjective optimization problems. J. Optim. Theory Appl. 122, 521–538 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Kellogg, R.B., Li, T.Y., Yorke, J.A.: A constructive proof of the Brouwer fixed-point theorem and computational results. SIAM J. Numer. Analysis 18, 473–483 (1976)

    Article  MathSciNet  Google Scholar 

  6. Chow, S.N., Mallet-Paret, J., Yorke, J.A.: Finding zeros of maps: Homotopy methods that are constructive with probability one. Math. Comput. 32, 887–899 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  7. Allgower, E.L., Georg, K.: Numerical Continuation Methods: An Introduction. Springer, Berlin (1990)

    MATH  Google Scholar 

  8. Garcia, C.B., Zangwill, W.I.: Pathways to Solutions, Fixed Points and Equilibria. Prentice-Hall, Englewood Cliffs (1981)

    MATH  Google Scholar 

  9. Megiddo, N.: Pathways to the optimal set in linear programming. In: Progress in Mathematical Programming, Interior Point and Related Methods, pp. 131–158. Springer, New York (1988)

    Google Scholar 

  10. Kojima, M., Mizuno, S., Yoshise, A.: A primal-dual interior point algorithm for linear programming. In: Megiddo, N. (ed.) Progress in Mathematical Programming, Interior Point and Related Methods, pp. 29–47. Springer, New York (1988)

    Google Scholar 

  11. McCormick, G.P.: The projective SUMT method for convex programming. Math. Oper. Res. 14, 203–223 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  12. Monteiro, R.D.C., Adler, I.: An extension of karmarkar type algorithm to a class of convex separable programming problems with global linear rate of convergence. Math. Oper. Res. 15, 408–422 (1990)

    MATH  MathSciNet  Google Scholar 

  13. Lin, Z.H., Yu, B., Feng, G.C.: A combined homotopy interior method for convex nonlinear programming. Appl. Math. Comput. 84, 93–211 (1997)

    Article  MathSciNet  Google Scholar 

  14. Lin, Z.H., Li, Y., Yu, B.: A combined homotopy interior point method for general nonlinear programming problems. Appl. Math. Comput. 80, 209–224 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  15. Lin, Z.H., Zhu, D.L., Sheng, Z.P.: Finding a minimal efficient solution of a convex multiobjective program. J. Optim. Theory Appl. 118, 587–600 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Di, S.: Classical optimality conditions under weaker assumptions. SIAM J. Optim. 6, 178–197 (1996)

    Article  MathSciNet  Google Scholar 

  17. Naber, G.L.: Topological Methods in Euclidean Spaces. Cambridge University Press, London (1980)

    MATH  Google Scholar 

  18. Zhang, Z.S.: Introduction to Differential Topology. Beijing University Press, Beijing (2002) (in Chinese)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Harbin Normal University, Harbin, 150080, China

    W. Song

  2. Department of Mathematics, Harbin College, Harbin, China

    G. M. Yao

Authors
  1. W. Song
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  2. G. M. Yao
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Correspondence to W. Song.

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Communicated by T. Rapcsák.

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Song, W., Yao, G.M. Homotopy Method for a General Multiobjective Programming Problem. J Optim Theory Appl 138, 139–153 (2008). https://doi.org/10.1007/s10957-008-9366-6

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  • DOI: https://doi.org/10.1007/s10957-008-9366-6

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