Research On Severval Nonlinear Problems Via Improvement Sets

The research on nonlinear problems including set-valued optimization and vector equilibrium problems plays an important role in many areas such as transportation,economic administration and optimal control.Especially,the concept of improvement sets is proposed and it offers an effective approach to unify exact and approximate solutions for set-valued optimization and equilibrium problems that attracts wide attention at home and abroad.In this paper,we mainly investigate optimality conditions for set-valued optimization and equilibrium problems via improvement sets in real ordered linear space without topological structure.The main contents are as follows:Firstly,we introduce the concept of globally proper efficient point via improvement sets in real ordered linear spaces.By use of the properties of algebraic interior and vector closure of sets,the relationships between globally proper efficiency via improvement sets and relevant concepts are discussed.Secondly,we consider two kinds of scalar optimization problems induced by set-valued optimization problems.A characterization of globally proper efficiency via improvement sets is given by the properties of the nonlinear scalar function via improvement sets and nearly E-subconvexlikeness of set-valued maps.Thirdly,Lagrange multiplier theorems of globally proper efficient solutions via improvement sets are established.We introduce the concept of globally proper saddle point via improvement sets in real ordered linear spaces and obtain the equivalent characterization of globally proper saddle point via improvement sets.Finally,we consider equilibrium problems and fixed point problems via improvement sets.Optimality conditions of solutions for equilibrium problems and common fixed point theorems via improvement sets in real ordered linear space are given.