Unbounded components in the solution sets of strictly quasiconcave vector maximization problems

摘要

Let (P) denote the vector maximization problem max{f(x) = (f_1(x),... f_m(x)):x ∈ 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.