The Proximal Point Algorithm For Solving Cone Constrained Optimization Problems

The proximal point algorithm is a kind of algorithm widely used in solving optimization problem,fixed point problem,maximal monotone operator and so on.It is not only closely related to many algorithms,but also can better explain and extend existing algorithms through the framework of the proximal point algorithm.Even for non-smooth augmented real-valued functions,convergence can be maintained under the proximal point algorithm.In recent years,combining the idea of the proximal point algorithm or proximal term with existing algorithms can improve the performance of the original algorithm to a certain extent.In this thesis,based on the application of proximal point algorithm in convex optimization problem,and with the help of Lagrange function,the optimization problem with cone constraint in finite dimensional space is transformed into minimax problem.In chapter 1,the main research background of proximal point algorithm in the optimization field is introduced,and then the research results and development of cone-constrained optimization problem are introduced.Finally,the basic knowledge and some basic conclusions related to this thesis are introduced.In chapter 2,the cone-constrained optimization problem is transformed into a minimax problem whose objective function is convex and concave.Then solving the original problem is equivalent to solving the minimax problem.It is proved that Lagrange function is convex and concave,that is about x is convex and about ? is concave.Finally,it is proved that TL(x,?)is maximal monotone operator.In chapter 3,optimality conditions are given on the basis of TL(x,?)is maximal monotone operators,that is,(0,0)? TL(x,y)if and only if(x,y)=arg min i max L.Based on the application of proximal point algorithm in minimax problem,the proximal point algorithm for solving cone-constrained optimization problem is obtained,and the convergence property and convergence rate are analyzed.