| Convexity and generalized convexity play a center role in mathematical economicis, engineering, management science, and optimization theory. Therefore, the research on convexity and generalized convexity is one of the most important aspects in mathematical programming. In this thesis, we mainly make further discuss the following several classes of generalized convex functions. First, The concept of B - C -semi-preinvex for vector-valued map is introduced, some properties concerning B-C-semi-preinvex for vector-valued map are obtained, and the relations between the local weak minimum and the global weak minimum for vectoroptimization problem (vp)(?) F(x) with B -C-semi-preinvex are discussed. Under thearcwise directionally differentiable condition, the equivalence relations between the vector optimization problem (vp) (?) F(x)and the vector variational-like inequality(vvli)find (x0∈K such that F'(x0;(?)(x,x0))(?)-intC ) are obtained. Secondly, suchdefinitions as D -τ-properly semi-prequasiinvex, D -τ-properly strongly semi-prequasiinvex, D-τ-properly strictly semi-prequasiinvex for vector-valued maps are introduced. The relations among D-τ-properly semi-prequasiinvex, D -τ-properly strongly semi-prequasiinvex, D -τ-properly strictly semi-prequasiinvex for vector-valued maps are discussed. Under some conditions, the fact that the local weak efficient solution of (VP) must be the global weak efficient solution of (VP) is proved. Finally, the concepts of generalized alpha-order C -preinvex set-valued function and generalized alpha-order invex set were introduced, from which the properties of the alpha-order contingent epiderivative was studied. Finally, we establish the necessary and sufficient optimality conditions for generalized alpha-order C -preinvex set-valued function in unconstrained set-valued optimization, the necessary condition for generalized alpha-order C -preinvex set-valued function in constrained set-valued optimization.
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