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 \(\mathbb{R}^{n}\). 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 \(f_{i} (i =1,\ldots, m)\) 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 \textit{E. U. Choo} and \textit{D. R. Atkis} [J. Optimization Theory Appl. 36, 203–220 (1982; Zbl 0452.90076)] 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 \(D\).