Unbounded Components in the Solution Sets of Strictly Quasiconcave Vector Maximization Problems |
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Authors: | T N Hoa N Q Huy T D Phuong N D Yen |
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Institution: | (1) Institute of Mathematics, Vietnamese Academy of Science and Technology, 18 Hoang Quoc Viet Road, 10307 Hanoi, Vietnam |
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Abstract: | Let (P) denote the vector maximization problem
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. |
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Keywords: | Strictly quasiconcave vector maximization problem Efficient solution set Weakly efficient solution set Unbounded component compactification procedure |
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