Taiwanese Journal of Mathematics

LAGRANGIAN DUALITY FOR VECTOR OPTIMIZATION PROBLEM WITH SET-VALUED MAPPINGS

Jian-Wen Peng and Xian Long

Full-text: Open access

Abstract

In this paper, by using a alternative theorem, we establish Lagrangian conditions and  duality results for set-valued vector optimization problems when the objective and constant are nearly cone-subconvexlike multifunctions in the sense of $E$-weak minimizer.

Article information

Source
Taiwanese J. Math., Volume 17, Number 1 (2013), 287-297.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499705888

Digital Object Identifier
doi:10.11650/tjm.17.2013.1935

Mathematical Reviews number (MathSciNet)
MR3028870

Zentralblatt MATH identifier
1290.90067

Subjects
Primary: 90C29: Multi-objective and goal programming 90C46: Optimality conditions, duality [See also 49N15]

Keywords
set-valued vector optimization problems Lagrangian duality alternative theorem $E$-weak minimizer nearly cone-subconvexlikeness

Citation

Peng, Jian-Wen; Long, Xian. LAGRANGIAN DUALITY FOR VECTOR OPTIMIZATION PROBLEM WITH SET-VALUED MAPPINGS. Taiwanese J. Math. 17 (2013), no. 1, 287--297. doi:10.11650/tjm.17.2013.1935. https://projecteuclid.org/euclid.twjm/1499705888


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References

  • H. W. Corley, Duality theory for maximizations with respect to cones, J. Math. Anal. Appl., 84 (1981), 560-568.
  • J. M. Borwein, Proper efficient points for maximizations with respect to cones, SIAM J. Control Optim., 15 (1977), 57-63.
  • M. Hayashi and H. Komiya, Perfect duality for convexlike programs, J. Optim. Theory Appl., 38 (1982), 179-189.
  • J. Jahn, Mathematical Vector Optimization in Partially Ordered Linear Spaces, Peter Lang, Frankfurt, Gemany, 1986.
  • B. D. Craven and V. Jeyakumar, Alternative and duality theorems with weakened convexity, Utilitas Mathematica, 31 (1987), 149-159.
  • V. Jeyakumar, W. Oettli and M. Natividad, A solvability theorem for a class of quasiconvex mapping with applications to optimization, J. Math. Anal. Appl., 179 (1993), 537-546.
  • T. Illes and G. Kassay, Theorems of the alternative and optimality conditions for convexlike and general convexlike programming, J. Optim. Theory Appl., 101 (1999), 243-257.
  • P. Q. Khanh and T. H. Nuong, On necessary and sufficient conditions in vector optimization, J. Optim. Theory Appl., 63 (1989), 391-413.
  • J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser, Basel, Switzerland, 1990.
  • K. Klein and A. C. Thompson, Theory of Correspondences, John Wiley, New York, 1984.
  • S. M. Robinson, Generalized equations and their solutions, part I: basic theory, Mathematical Programming Study, 10 (1979), 128-141.
  • H. W. Corley, Existence and Lagrangian duality for maximizations of set-valued functions, J. Optim. Theory Appl., 54 (1987), 489-501.
  • D. T. Luc, Theory of Vector Optimization, Springer-Verlag, Berlin, Germany, 1989.
  • D. Bhatia and A. Mehra, Lagrangian duality of preinvex set-valued functions, J. Math. Anal. Appl., 214 (1997), 599-612.
  • Z. F. Li and G. Y. Chen, Lagrangian multipliers, saddle points and duality in vector optimization with set-valued maps, J. Math. Anal. Appl., 215 (1997), 297-316.
  • W. D. Rong and Y. N. Wu, $\varepsilon$-Weak minimal solution of vector optimization problems with set-valued maps, J. Optim. Theory Appl., 106 (2000), 569-579.
  • W. Song, Lagrangian duality for minimization of nonconvex multifunctions, J. Math. Anal. Appl., 201 (1996), 212-225.
  • X. M. Yang, D. Li and S. Y. Wang, Near-subconvexlikeness in vector optimization with set-valued functions, J. Optim. Theory Appl., 110 (2001), 413-427.
  • P. H. Sach, Nearly subconvexlikeness set-valued maps and vector optimization problems, J. Optim. Theory Appl., 119 (2003), 335-356.
  • J. W. Peng and X. M. Yang, $e$-Proper efficient solutions of vector optimization problems with set-valued maps, Appl. Math. J. Chinese Univ. Ser. A., 16 (2001), 486-492. (in Chinese)
  • Y. W. Huang, Optimality conditions for vector optimization with set-valued maps, Bull. Austra. Math. Soc., 66 (2002), 317-330.
  • W. W. Breckner and G. Kassay, A systematization of convexity concepts for sets and functions, Journal of Convex Analysis, 4 (1997), 109-127.