Some Research On Vector Optimization And Scalarization Functions

In this thesis,the relationships and applications of two nonlinear scalarization functions are discussed.Based on the image space analysis,optimality conditions and penalty functions for constrained vector(set-valued)optimization problems,and the gap functions and error bounds for vector equilibrium problems are studied.The thesis consists of the following six chapters:In Chapter 1,the background,academic meaning and the domestic and overseas research of optimization problems are briefly introduced firstly.Then the researches on vector optimization problems,vector equilibrium problems,scalarization functions and image space analysis are reviewed.Finally,the motivations and the main content of this thesis are presented.In Chapter 2,some notations,definitions,and basic properties are given,which including the definitions and properties of two nonlinear scalarization functions,tangent cones and generalized derivatives for set-valued maps,and the separation functions in image space analysis.In Chapter 3,the relationships between Gerstewitz function and oriented distance function and their applications are discussed.The monotonic properties of the oriented distance functional are obtained when the associated set is neither a cone nor a convex set.Then,several relationships between the two nonlinear scalarization functions are analyzed,and an equivalent relation is proved under a suitable norm assumption.By using these results,the two functions are applied in the characterizations of solution sets for the vector optimization problem and vector equilibrium problem.In Chapter 4,by virtue of the image space analysis,constrained vector(set-valued)optimization problem are considered for three cases.Firstly,by means of some suitable separation functions,the Lagrange type optimality conditions and characterizations of vector optimization problem with non-cone constraints are investigated under convex and nonconvex cases.Then,with the aid of the vector type nonlinear separation functions,we construct the vector penalty function for vector optimization problem with non-cone constraints and derive the penalization results.Secondly,the characterization of the Benson proper efficiency is established by using a suitable separation of two sets in image space.Then,the relations among the Benson proper efficiency,image regularity condition and regular separation are discussed.Moreover,we use a generalized Lagrangian function to investigate the Lagrange type sufficient and necessary optimality conditions for vector optimization problem without convexity assumption.Thirdly,we establish the sufficient and necessary optimality conditions for set-valued optimization problem with cone constraints in terms of tangent cone of extended image set.Simultaneously,some weaker regularity conditions are gotten by using the tangent cone of extended image set.Combining with the relations between different generalized derivatives of set-valued maps and extended image,Karush-Kuhn-Tucker sufficient and necessary optimality conditions for set-valued optimization problem are derived.In Chapter 5,gap functions and error bounds for vector equilibrium problems are investigated by using the image space analysis.On one hand,gap functions for vector equilibrium problem are constructed by virtue of a class of regular weak separation functions.Then,under strong monotonicity and some suitable assumptions,the error bound is obtained in terms of the gap function when the solution set is a set.On the other hand,with the help of the general regular weak separation functions,together with some suitable assumptions,we obtain some unify gap functions for vector equilibrium problems.Subsequently,using these gap functions,some error bounds are established under some monotonicity and appropriate conditions.In Chapter 6,we simply summarize the thesis,as well present some problems,which wait for being thought and following researched.