Some Research On Separation Functions Of Image Space Analysis For Semidefinite Programming

In this paper,we mainly study the properties of functions used in image space analysis for semidefinite programming.Image space analysis(ISA)was proposed by Giannessi in 1979 to study constrained optimization from a geometric point of view.ISA has attracted the attention of many scholars at home and abroad,and has been further developed.Based on the theoretical framework established by giannessi,this paper tentatively uses image space analysis to solve semidefinite programming problems,and has achieved some results.In this paper,we first introduce some important concepts,such as separation function and directed distance function,which are used in the theory of image space analysis.Then,this paper gives several kinds of separation functions,studies their scope of application,discusses the relationship between the function and the optimal solution,and proves them.The main work of this paper is as follows:1.In Chapter 2,the basic theory of ISA is introduced,including the definition of several sets commonly used,the relationship between the optimal conditions of the original problem and the sets in the image space.2.In Chapter 3,according to the form of constraint function in semidefinite programming problem,we propose two kinds of separation function corresponding to semidefinite programming problem,and give some forms of separation function,find the corresponding applicable conditions of these separation functions,including the parameters involved in the function and the specific properties of the function itself.Then under these conditions,it is proved that this kind of function satisfies the properties of the separation functions.Finally,for the Lagrangian function of the original problem,the equality and inequality relations related to the Lagrangian function are discussed,and the proof process is deduced by the optimization theory method.