Advertisement

Journal of Global Optimization

, Volume 37, Issue 1, pp 1–10 | Cite as

Unbounded Components in the Solution Sets of Strictly Quasiconcave Vector Maximization Problems

  • T. N. Hoa
  • N. Q. Huy
  • T. D. Phuong
  • N. D. YenEmail author
Original Paper

Abstract

Let (P) denote the vector maximization problem
$$\max\{f(x)=\big(f_1(x),\ldots,f_m(x)\big){:}\,x\in D\},$$
where the objective functions f i are strictly quasiconcave and continuous on the feasible domain D, which is a closed and convex subset of R n . We prove that if the efficient solution set E(P) of (P) is closed, disconnected, and it has finitely many (connected) components, then all the components are unbounded. A similar fact is also valid for the weakly efficient solution set E w (P) of (P). Especially, if f i (i=1,...,m) are linear fractional functions and D is a polyhedral convex set, then each component of E w (P) must be unbounded whenever E w (P) is disconnected. From the results and a result of Choo and Atkins [J. Optim. Theory Appl. 36, 203–220 (1982.)] it follows that the number of components in the efficient solution set of a bicriteria linear fractional vector optimization problem cannot exceed the number of unbounded pseudo-faces of D.

Keywords

Strictly quasiconcave vector maximization problem Efficient solution set Weakly efficient solution set Unbounded component compactification procedure 

2000 Mathematics Subject Classifications

90C29 90C26 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Benoist J. (1998). Connectedness of the efficient set for strictly quasiconcave sets. J. Optim. Theory Appl. 96: 627–654CrossRefGoogle Scholar
  2. Choo E.U., Atkins D.R. (1982). Bicriteria linear fractional programming. J. Optim. Theory Appl. 36: 203–220CrossRefGoogle Scholar
  3. Choo E.U., Atkins D.R. (1983). Connectedness in multiple linear fractional programming. Manage. Sci. 29: 250–255CrossRefGoogle Scholar
  4. Daniilidis A., Hadjisavvas N., Schaible S. (1997). Connectedness of the efficient set for three-objective quasiconcave maximization problems. J. Optim. Theory Appl. 93: 517–524CrossRefGoogle Scholar
  5. Hoa T.N., Phuong T.D., Yen N.D. (2005a). Linear fractional vector optimization problems with many components in the solution sets. J. Ind. Manage. Optim. 1: 477–486Google Scholar
  6. Hoa T.N., Phuong T.D., Yen N.D. (2005b). On the parametric affine variational inequality approach to linear fractional vector optimization problems. Vietnam J. Math. 33: 477–489Google Scholar
  7. Huy N.Q., Yen N.D. (2004). Remarks on a conjecture of J. Benoist. Nonlinear Anal. Forum 9: 109–117Google Scholar
  8. Huy N.Q., Yen N.D. (2005). Contractibility of the solution sets in strictly quasiconcave vector maximization on noncompact domains. J. Optim. Theory Appl. 124: 615–635CrossRefGoogle Scholar
  9. Luc D.T. (1987). Connectedness of the efficient set in quasiconcave vector maximization. J. Math. Anal. Appl. 122: 346–354CrossRefGoogle Scholar
  10. Luc D.T. (1989). Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, vol. 319. Springer-Verlag, BerlinGoogle Scholar
  11. Schaible S. (1983). Bicriteria quasiconcave programs. Cahiers du Centre d’Etudes de Recherche Opérationnelle 25: 93–101Google Scholar
  12. Warburton A.R. (1983). Quasiconcave vector maximization: connectedness of the set of pareto-optimal and weak pareto-optimal alternatives. J. Optim. Theory Appl. 40: 537–557CrossRefGoogle Scholar
  13. Yen N.D., Phuong T.D. (2000). Connectedness and stability of the solution sets in linear fractional vector optimization problems. In: Giannessi F. (eds) Vector Variational Inequalities and Vector Equilibria. Kluwer Academic Publishers, Dordrecht, pp. 479–489Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • T. N. Hoa
    • 1
  • N. Q. Huy
    • 1
  • T. D. Phuong
    • 1
  • N. D. Yen
    • 1
    Email author
  1. 1.Institute of MathematicsVietnamese Academy of Science and TechnologyHanoiVietnam

Personalised recommendations