Numerical Algorithms

, Volume 69, Issue 4, pp 819–838 | Cite as

A Barzilai-Borwein type method for minimizing composite functions

  • Yakui HuangEmail author
  • Hongwei Liu
Original Paper


In this paper, we propose a Barzilai-Borwein (BB) type method for minimizing the sum of a smooth function and a convex but possibly nonsmooth function. At each iteration, our method minimizes an approximation function of the objective and takes the difference between the minimizer and the current iteration as the search direction. A nonmonotone strategy is employed for determining the step length to accelerate the convergence process. We show convergence of our method to a stationary point for nonconvex functions. Consequently, when the objective function is convex, the proposed method converges to a global solution. We establish sublinear convergence of our method when the objective function is convex. Moreover, when the objective function is strongly convex the convergence rate is R-linear. Preliminary numerical experiments show that the proposed method is promising.


Barzilai-Borwein method Linear convergence Nonmonotone 2- 1 minimization 


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXidian UniversityXi’anPeople’s Republic of China

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