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Journal of Applied Mathematics
Volume 2011, Article ID 464832, 21 pages
http://dx.doi.org/10.1155/2011/464832
Research Article

Generating Efficient Outcome Points for Convex Multiobjective Programming Problems and Its Application to Convex Multiplicative Programming

Faculty of Applied Mathematics and Informatics, Hanoi University of Technology, Dai Co Viet, Hai Ba Trung, Hanoi, Vietnam

Received 30 November 2010; Revised 7 April 2011; Accepted 28 May 2011

Academic Editor: Ya Ping Fang

Copyright © 2011 Le Quang Thuy et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. H. P. Benson, “An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem,” Journal of Global Optimization, vol. 13, no. 1, pp. 1–24, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. J. Jahn, Vector Optimization: Theory, Applications and Extensions, Springer, Berlin, Germany, 2004.
  3. M. T. Tabucanon, “Multiobjective programming for industrial engineers,” in Mathematical Programming for Industrial Engineers, M. Avriel et al., Ed., vol. 20 of Indust. Engrg., pp. 487–542, Marcel Dekker, New York, NY, USA, 1996. View at Google Scholar · View at Zentralblatt MATH
  4. P. Armand, “Finding all maximal efficient faces in multiobjective linear programming,” Mathematical Programming, vol. 61, no. 3, pp. 357–375, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. H. P. Benson and E. Sun, “Outcome space partition of the weight set in multiobjective linear programming,” Journal of Optimization Theory and Applications, vol. 105, no. 1, pp. 17–36, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. A. Chinchuluun and P. M. Pardalos, “A survey of recent developments in multiobjective optimization,” Annals of Operations Research, vol. 154, pp. 29–50, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. M. Ehrgott, L. Shao, and A. Schöbel, “An approximation algorithm for convex multi-objective programming problems,” Journal of Global Optimization, vol. 50, no. 3, pp. 397–416, 2011. View at Publisher · View at Google Scholar
  8. N. T. B. Kim and D. T. Luc, “Normal cones to a polyhedral convex set and generating efficient faces in linear multiobjective programming,” Acta Mathematica Vietnamica, vol. 25, no. 1, pp. 101–124, 2000. View at Google Scholar · View at Zentralblatt MATH
  9. N. T. B. Kim and D. T. Luc, “Normal cone method in solving linear multiobjective problems,” Journal of Statistics & Management Systems, vol. 5, no. 1-3, pp. 341–358, 2002. View at Google Scholar · View at Zentralblatt MATH
  10. N. T. B. Kim, N. T. Thien, and L. Q. Thuy, “Generating all efficient extreme solutions in multiple objective linear programming problem and its application to multiplicative programming,” East-West Journal of Mathematics, vol. 10, no. 1, pp. 1–14, 2008. View at Google Scholar · View at Zentralblatt MATH
  11. 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. 3, pp. 587–600, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. D. T. Luc, T. Q. Phong, and M. Volle, “Scalarizing functions for generating the weakly efficient solution set in convex multiobjective problems,” SIAM Journal on Optimization, vol. 15, no. 4, pp. 987–1001, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. W. Song and G. M. Yao, “Homotopy method for a general multiobjective programming problem,” Journal of Optimization Theory and Applications, vol. 138, no. 1, pp. 139–153, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. H. P. Benson, “An outcome space branch and bound-outer approximation algorithm for convex multiplicative programming,” Journal of Global Optimization, vol. 15, no. 4, pp. 315–342, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. B. Jaumard, C. Meyer, and H. Tuy, “Generalized convex multiplicative programming via quasiconcave minimization,” Journal of Global Optimization, vol. 10, no. 3, pp. 229–256, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. N. T. B. Kim, “Finite algorithm for minimizing the product of two linear functions over a polyhedron,” Journal of Industrial and Management Optimization, vol. 3, no. 3, pp. 481–487, 2007. View at Google Scholar · View at Zentralblatt MATH
  17. N. T. B. Kim, N. T. L. Trang, and T. T. H. Yen, “Outcome-space outer approximation algorithm for linear multiplicative programming,” East-West Journal of Mathematics, vol. 9, no. 1, pp. 81–98, 2007. View at Google Scholar · View at Zentralblatt MATH
  18. L. D. Muu and B. T. Tam, “Minimizing the sum of a convex function and the product of two affine functions over a convex set,” Optimization, vol. 24, no. 1-2, pp. 57–62, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. N. V. Thoai, “A global optimization approach for solving the convex multiplicative programming problem,” Journal of Global Optimization, vol. 1, no. 4, pp. 341–357, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. H. Tuy and N. D. Nghia, “Reverse polyblock approximation for generalized multiplicative/fractional programming,” Vietnam Journal of Mathematics, vol. 31, no. 4, pp. 391–402, 2003. View at Google Scholar · View at Zentralblatt MATH
  21. P. L. Yu, Multiple-Criteria Decision Making, vol. 30 of Mathematical Concepts and Methods in Science and Engineering, Plenum Press, New York, NY, USA, 1985.
  22. R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, no. 28, Princeton University Press, Princeton, NJ, USA, 1970.
  23. D. T. Luc, Theory of Vector Optimization, vol. 319 of Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, 1989.
  24. A. M. Geoffrion, “Proper efficiency and the theory of vector maximization,” Journal of Mathematical Analysis and Applications, vol. 22, pp. 618–630, 1968. View at Publisher · View at Google Scholar · View at Zentralblatt MATH