Weak efficiency in vector optimization using a closure of algebraic type under cone-convexlikeness

We introduce a new concept of algebraic type of closure in real linear spaces which is called vector closure. The properties of this are nearer to the topological closure than the algebraic closure. Through this closure we introduce new concepts of generalized convexlikeness, with which it is possible to characterize the weakly efficient solutions in vector optimization (with and without constraints) through scalarization, multiplier rule and saddle-point theorems.