Linear fractional vector optimization problems with many components in the solution sets
Tran N. Hoa, Tạ Duy Phượng, Nguyen Dong Yen · Journal of Industrial and Management Optimization · 2005
Linear fractional vector optimization (LFVO) problems form a special class of nonconvex multiobjective optimization problems which has a significant role both in the management science and in the theory of vector optimization. Up to now, only LFVO problems with at most two connected components in the solution sets have been discussed in the literature. We propose some examples of LFVO problems with three or more connected components in the solution sets. It is proved that for any integer $m$ there exist LFVO problems with $m$ objective criteria whose solution sets have exactly $m$ connected components. Besides, we have solved the conjecture saying that $\chi(E(\mbox{P}))\leq \min\{m,\mbox{dim}0^+D+1\},$ where $\chi(E(\mbox{P}))$ is the number of connected components in the efficient solution set of a LFVO problem $(\mbox{P})$, $m$ is the number of the objective criteria of $(\mbox{P})$, and $\mbox{dim}0^+D$ is the dimension of the recession cone $0^+D$ of the feasible domain $D$ of $(\mbox{P})$. These new facts are useful for analyzing the practical problems which can be modeled as quasiconcave vector maximization problems in general, and as LFVO problems on unbounded feasible domains in particular.