Research On Two Types Of Barzilai-Borwein Algorithms

Barzilai-Borwein(BB)method,which is a kind of gradient method with the step containing the second-order information,was introduced by Barzilai and Borwein in 1988.Until now,many experts and scholars are still doing researches in this field because of the simple construction,the small storage and the high efficiency of BB method.We mainly introduce two kinds of BB method with new step size.There are four chapters in this article.In Chapter 1,we summarize the related knowledge of BB algorithm at home and abroad,as well as its applications in real life.In Chapter 2,we propose another new generalized BB step for the strictly convex quadratic function problems,and describe concrete steps of the algorithm at the same time.Then,we analyze the convergence of the new extended BB algorithm and provide the new extended BB algorithm with the nonmonotone line search technique.Finally,we show the effectiveness of these algorithms by the related experiments.In Chapter 3,we give a further analysis of Cauchy-Barzilai-Borwein algorithm,and propose two new Cauchy-Barzilai-Borwein algorithms as well.Then,we give the convergence analysis of these two algorithms under certain assumptions.Finally,the validity of the algorithms can be verified by some numerical experiments.In Chapter 4,We give a summary of the research contents in this paper,and put forward the future research problems which needs to be resolved.