The Barzilai-Borwein (BB) method is a popular and efficient tool for solving large-scale unconstrained optimization problems. Its search direction is the same as for the steepest descent (Cauchy) method, but its stepsize rule is different. Owing to this, it converges much faster than the Cauchy method. A feature of the BB method is that it may generate too long steps, which throw the iterates too far away from the solution. Moreover, it may not converge, even when the objective function is strongly convex. In this paper, a stabilization technique is introduced. It consists in bounding the distance between each pair of successive iterates, which often allows for decreasing the number of BB iterations. When the BB method does not converge, our simple modification of this method makes it convergent. Under suitable assumptions, we prove its global convergence, despite the fact that no line search is involved, and only gradient values are used. Since the number of stabilization steps is proved to be finite, the stabilized version inherits the fast local convergence of the BB method. The presented results of extensive numerical experiments show that our stabilization technique often allows the BB method to solve problems in a fewer iterations, or even to solve problems where the latter fails.
Barzilai and Borwein conjugate gradient method equipped with a non-monotone line search technique and its application on non-negative matrix factorization
A variable metric mini-batch proximal stochastic recursive gradient algorithm with diagonal Barzilai-Borwein stepsize
Hebei University of Technology（河北工业大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；National University of Singapore（新加坡国立大学）
Barzilai-Borwein Step Size for Stochastic Gradient Descent
The Chinese University of Hong Kong（香港中文大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；The University of Hong Kong（香港大学）
Adaptive Bregman Proximal Stochastic Gradient with a Stabilized Barzilai--Borwein Step Size
University of Utah（犹他大学）；Beijing Normal-Hong Kong Baptist University（北京师范大学-香港浸会大学联合国际学院）；Guangzhou Nanfang College（广州南方学院）；The Chinese University of Hong Kong（香港中文大学）
Stochastic Non-convex Ordinal Embedding with Stabilized Barzilai-Borwein Step Size
Institute of Information Engineering, Chinese Academy of Sciences（中国科学院信息工程研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Jiangxi Normal University（江西师范大学）；Hong Kong University of Science and Technology（香港科技大学）；Tencent AI Lab（腾讯人工智能实验室）
$R$-Linear Convergence of Barzilai-Borwein Methods on Strictly Convex Quadratics via Closed-Cone Homogeneous Dynamics
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
Nonmonotone Barzilai-Borwein Gradient Algorithm for $\ell_1$-Regularized Nonsmooth Minimization in Compressive Sensing
Institute of Applied Mathematics, College of Mathematics and Information Science, Henan University（河南大学数学与信息科学学院应用数学研究所）；National Center for Theoretical Sciences (South), National Cheng Kung University（台湾成功大学南方理论科学中心）；Department of Applied Mathematics, Hong Kong Polytechnic University（香港理工大学应用数学系）
The Sharp Worst-Case Asymptotic Rate of the Barzilai--Borwein Method in $\mathbb R^d$ and Hilbert Spaces
State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算国家重点实验室）；University of Chinese Academy of Sciences（中国科学院大学）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
Lyapunov Functions and $R$-Linear Convergence for Quadratic Barzilai-Borwein Dynamics
State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算国家重点实验室）；University of Chinese Academy of Sciences（中国科学院大学）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
Adaptation and Development of Super Schemes for Unconstrained Optimization Problems
Institute of Mathematics and Digital Technology, Mongolian Academy of Sciences（蒙古国科学院数学与数字技术研究所）；School of Applied Sciences, Mongolian University of Science and Technology（蒙古科技大学应用科学学院）；Faculty of Mechanical and Electrical Engineering, German-Mongolian Institute for Resources and Technology（德蒙资源与技术学院机械与电气工程学院）
