Generalized Splitting Algorithm For Solving The Bilevel Convex Optimization Problem

The dissertation contains two parts.The first part constructs a generalized Forward-Backward splitting algorithm for solving bilevel convex optimization problem.The second part constructs an improved Forward-Backward splitting algorithm for solving the inner layer problem of the bilevel convex optimization problem.The first part constructs a generalized Forward-Backward splitting algorithm for solving bilevel convex optimization problem.The Forward-Backward splitting algorithm is an effective algorithm for solving the convex optimization problem.It can easily achieve parallel computing,especially suitable for large-scale optimization problem.In the case of bilevel convex optimization problem,Sabach and Shimrit propose a Bi G-SAM to solve it,and they proves its convergence.In this dissertation,we propose the generalized Forward-Backward algorithm to solve the bilevel convex optimization problem.Compared with the classical Forward-Backward splitting algorithm,it is more flexible in parameter selection.By virtue of the property of nonexpansive operator,the convergence of the algorithm is given.The second part constructs an improved Forward-Backward splitting algorithm for solving the inner layer problem of the bilevel convex optimization problem.A classical modification of Forward-Backward method was proposed by Tseng,provided that both the forward and the backward operators are monotone and that the backward operator is Lipschitz continuous.The algorithm proposed here improves Tseng's method in some instances.The first and main part of our approach,contains an explicit Armijo-type search in the spirit of the extragradient-like methods for variational inequalities.During the iteration process,the search performs only one calculation of the forward-backward operator,in each tentative of the step.This achieves a considerable computational saving when the forward-backward operator is computationally expensive.The second part of the scheme consists in special projection steps.The convergence analysis of the proposed scheme is given assuming monotonicity on both operators,without Lipschitz continuity assumption on the backward operator.