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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Hoa, T.N., Huy, N.Q., Phuong, T.D. et al. Unbounded Components in the Solution Sets of Strictly Quasiconcave Vector Maximization Problems. J Glob Optim 37, 1–10 (2007). https://doi.org/10.1007/s10898-006-9032-1
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DOI: https://doi.org/10.1007/s10898-006-9032-1
Keywords
- Strictly quasiconcave vector maximization problem
- Efficient solution set
- Weakly efficient solution set
- Unbounded component
- compactification procedure