The Ic-Cone-Convexlikeness And Its Applications In Vector Optimization Problems With Set-Valued Maps

Recently, a new generalized convexity for set-valued maps called ic-cone-convexli-keness was introduced by P. H. Sach in Ref. [1]. In this thesis, we research this generalizedconvexity and its applications in vector optimization problems with set-valued maps.Firstly, we discuss the relationships between ic-cone-convexlikeness and other general-ized convexities. Secondly, we deal with vector optimization problem (VP) with ic-cone-convexlike set-valued maps. Theε-weakly efficient solution,ε-properly efficient solutionand c-supper efficient solution of this problem are discussed. We obtain the scalarizationresults, Lagrangian theorems and duality theorems for this problem. Finally, we pointout that theε-strong efficient solutions of (VP) can be discussed under the assumptionof some generalized convexities which are weaker than cone-convexlikeness, and a neces-sary condition is established in the sense of e-strong efficiency under the assumption ofic-cone-convexlikeness for set-valued maps.