Study On Characterization Of Approximate Solutions In Vector Optimization Problems

The research on characterizations of approximate solutions in vector optimizationproblems is an important aspect in the field of the theory and applications of vectoroptimization. In this paper, we mainly establish the Lagrange multiplier theorem forε-weakly efcient solution in vector optimization problems with set-valued maps withinequality-equality constraints system. Secondly, we propose a new approximate properefcient solution of vector optimization problems with set-valued maps with inequalityconstraints, and establish the linear scalarization theorem under the assumption of n-early E-subconvexlikeness, and also establish the Lagrange multiplier theorem. Thirdly,we give an equivalent characterization of E-Benson proper efcient solution for vectoroptimization problems.In chapter2, we consider a class of vector optimization problems with set-valuedmaps with inequality-equality constraints system, and propose the concept of ε-weaklyefcient solution. Furthermore, under the assumption of nearly cone-subconvexlikeness,we establish a necessary condition and an equivalent condition for ε-weakly efcient so-lution for vector optimization problems with set-valued maps by using an alternativetheorem. Then, we also establish the Lagrange multiplier theorem of ε-weakly efcientsolution for vector optimization problems with set-valued maps.In chapter3, based on the ideas of the Benson-proper efciency and E-Benson properefciency, we proposed a kind of new concept of approximate proper efciency, named asgeneralized E-Benson proper efciency for vector optimization problems with set-valuedmaps. This kind of new concept of approximate proper efciency contains some knownapproximate proper efciency as its special case. Moreover, we also established a linearscalarization theorem of generalized E-Benson proper efcient solution under the assump-tion of nearly E-subconvexlikeness. At last, we also give a Lagrange multiplier theorem.In chapter4, we mainly give an equivalent characterization of E-Benson properefciency in vector optimization. This type of characterization is a generalization of the classical equivalent relation of Benson proper efciency and Geofrion proper efciency infinite dimensional space with natural order.