Nechay I. On solvability and regularization of nonscalar optimization problems in Banach spaces

The object is problems of vector optimization. The aim is to study the problem of solvability for vector optimization problems in Banach spaces for which purpose map is not quasi-lower semi-continuous. The methods of functional analysis, nonlinear and convex analysis, theory of semi-ordered vector spaces have been used. The new notion of lower semi-continuity of vector-valued mappings is introduced. It is shown that the class of that vector-valued mappings contains the class of quasi-lower semi-continuous mappings. The sufficient conditions of the existence of efficient solutions to the corresponding vector optimization problems have been obtained. It was shown that using the approach of quasi-lower semi-continuous regularization can produce the situation when the properties of regularized problem are different from the properties of the corresponding regularized scalar problem. The concept of regularization of vector optimization problems have been proposed.