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

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.