The Optimality Condition Of The Nonconvex Set-valued Optimization Problem

Set-valued optimization theory is closely related to many branches of modern mathematics, and has an important role in the field of variational calculus, differ-ential calculus, and the optimal control, etc. The study of set-valued optimization theory has important theoretical significance and practical value since it involves many mathematical subjects, such as convex analysis, nonlinear functional analysis and nonsmooth analysis, etc. The optimality condition under various solutions of set-valued optimization problems is an important part of the theory of set-valued optimization, which is the important foundation of modern optimization algorithm. The optimality conditions of set-valued optimization problems and the theory of the structure of the solution set play an important role in the theory of set-valued optimization.The research of the optimization conditions has attracted extensive attention of researchers at home and abroad. In recent years, many researchers study all kinds of effective solution of the optimization conditions of and obtained some useful re-sults. DT (Difference of two set-valued maps) includes many nonconvex mappings, and many nonconvex optimization problems can be represented as DT optimization problems. Recently, some researchers established necessary or sufficient conditions of the existence of weak efficient solutions of the DT set-valued optimizations in terms of the sub differential of set-valued maps. Sharp minimal plays a very impor-tant application in convergence analysis and stability analysis of some algorithms. Many researchers established necessary or sufficient conditions of the existence of sharp minimal in terms of the subdifferential of set-valued maps and normal cones. Very recently, several authors studied some property of the Greenberg-Pierskalla subdifferential of numerical functions and its applications.In this thesis, we consider optimization conditions and sharp minimal of DT set-valued optimization problems with constraints and discussed the Greenberg- Pierskalla subdifferential of set-valued mappings. Firstly, we establish necessary conditions and sufficient conditions of the existence of ε-weak efficient solutions of the DT set-valued optimizations in terms of the ε-strong subdifferential and the ε-weak subdifferential of set-valued maps. Then we establish necessary and sufficient conditions of the existence of sharp minimal in terms of the strong sub differen-tial weak subdifferential of set-valued maps and normal cones. Finally we define the Greenberg-Pierskalla subdifferential of set-valued mappings, give an existence theorem of the Greenberg-Pierskalla subdifferential, and give a Fermat rule of set-valued optimization problems in terms of the Greenberg-Pierskalla subdifferential.