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


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.


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

2000 Mathematics Subject Classifications

90C29 90C26 


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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

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