Study On Characterizations Of Several Kinds Of Solutions In Vector Optimization Problems

Vector optimization theory and methods are widely used in the fields of economic planning, financial investment, engineering design, transportation System, environmen-tal conservation, decision science, etc. As one of the main research fields in vector optimization theory and applications, many researches on efficient solutions, weakly ef-ficient solutions and properly efficient solutions of vector optimization has been obtained. These results mainly concentrate upon the relationships between different solutions, the properties and existence of the solutions, the optimality conditions, duality theorems and saddle theorems, well-posedness of the solution, the characters and structure of the so-lution set such as compactness, connectedness, convergence, property of denseness and how to solve and analyze the solution of vector optimization, and so on. Until now, there are many different formals definitions of exact and approximate solutions (includ-ing various properly efficiency and approximate properly efficiency). In recent years, some new definitions especially the unified definitions on the solutions are proposed and studied continuously. Therefore, how to unify the solutions of the vector optimization and how to study the properties of them such as existence of the solutions, scalarization theorems, optimality conditions, denseness and stability of the solution set et al. under a general condition have very important significance. Currently, researches on the fields internationally are few.In the thesis, we propose the efficiency, weakly efficiency and super efficiency based on Assumption (A) and strict efficiency based on improvement set respectively. More-over, we discuss the relationships with the various classic efficient, properly efficient solutions and other unified solutions. Then, we study the characters of the solutions proposed under the unified framework. We mainly focus on the following three aspects here:(1). establish ing the scalarization theorems for several kinds of unified solutions by Linear functional and by Nonlinear scalarization functions; (2). presenting the E-strict efficiency based on improvement set and study its characterizations, the relationship with strict efficiency, approximate strict efficiency, E-super efficiency and E-Benson properly efficiency and their scalarization theorems and optimality conditions; (3). proposing the S-efficiency, weakly S-efficiency and S-super efficiency based on Assumption (A) and study their uniformity, scalarization theorems, Lagrange multiplier theorems under the assumption of S-subconvexlikeness. This thesis include the main achievements as fol-lows:First of all, we mainly study the characterizations of several kinds of unified solution introduced previously in the literatures by three types of nonlinear scalarization func-tion. We characterize the new (C, ε)-(weakly efficient solution proposed by Gutierrez et al. based on co-radiant set which unified some well-known concepts by two kinds of nonlinear scalarization function——Δ function and Gerstewitz function. For fur-ther study, some nonlinear scalarization characterizations of E-optimal point, weakly E-optimal point and E-Benson proper efficient point proposed via improvement set are established, respectively by a new monotonously increasing sublinear function based on Bishop-Phelps cone. These results improve and generalize the conclusions in the refer-ences. As a special case, the scalarization of Benson proper efficient point is also given. Furthermore, some examples are given to illustrate our main results.Secondly, we propose a kind of unified strict efficiency named E-strict efficiency via improvement sets for vector optimization problems and study the properties of E-strict efficiency. Firstly, we give the definition of E-strict efficiency based on improvement set proposed by Chicco and Gutierrez. Then, we study the characterizations of E-strict efficiency. This kind of efficiency is shown to have many desirable properties and be an extension of the classical strict efficiency and ε-strict efficiency. Moreover, we also discuss some relations with E-Benson proper efficiency and E-super efficiency and estab-lish the scalarization theorems and Lagrange multiplier theorems for E-strict efficiency under nearly E-subconvexlikeness proposed by Zhao et al. In the end, we obtain a non-linear scalarization theorem for E-strictly efficient solution by the nonlinear function--Gerstewitz function and the corresponding nonconvex separation theorem.Then, we propose S-(weakly)efficiency and S-super efficiency of vector optimiza-tion problems with set-valued maps via Assumption (A) which proposed by Flores-Bazan and Hernandez and study the properties of the solutions. Firstly, we propose the S-(weakly)efficiency and S-super efficiency of vector optimization problems based on the super efficiency and ε-super efficiency and discuss the relationships between (weakly)efficiency, E-(weakly)efficiency, super efficiency and ε-super efficiency. Fur- thermore, we also introduce the concept of nearly S-subconvexlikeness via Assump-tion (A) and discuss some relations with other generalized convexity which unifies E-subconvexlikeness, nearly E-subconvexlikeness and classical cone-subconvexlikeness in the literatures. Moreover, we obtain an alternative theorem under the assumption that the set-valued maps with S-subconvexlikeness. Also, under the assumption of S-subconvexlikeness, we establish scalarization theorems and Lagrange multiplier theo-rems for weakly S-efficient solution and D-super efficienct solution. At last, we estab-lished a nonlinear scalarization for S-super efficient solution by the nonlinear function--Gerstewitz function and the corresponding nonconvex separation theorem.Finally, we summarize the main results in this thesis and also give several open questions which are worthy of further researches associated with the main research in this thesis.