Hybrid Gradient Projection Algorithm Splitting Algorithm For Solving Convex Optimization Problems

Optimization theory and method is an important branch of mathematics and a very young discipline with strong application,it formulates the actual problem with mathematical language and obtains an abstract mathematical problem,and then designs a suitable algorithm for it,analyzes the performance of the algorithm and verifies the rationality and correctness of the mathematical model.This dissertation mainly studies the algorithm for solving constraint convex optimization problem.This dissertation,consisting of four parts,is organized as follows:In chapter 1,we introduce the research background and the situation at home and abroad,and main contents of this dissertation.In chapter 2,we recall some related concepts and conclusions which will be used in this dissertation.In chapter 3,we introduce a hybrid gradient projection algorithm of type Mann-Halpern to solve constrained convex minimization problems,and prove the strong convergence of this algorithm under some suitable conditions.In chapter 4,we study the Forward-Backward,Douglas-Rachford,and Backward-Backward splitting algorithm for solving bilevel optimization problems,and we prove the strong convergence of those algorithms under some suitable conditions.Since a variational inequality can be written as the inclusion of the sum of two operators,as an application,we apply the algorithm to study the bilevel optimization problem with the constrained of variational inequality,and give its convergence.The results improve and extend the research of Sabach and Shimrit.