Computational Optimization and Applications

, Volume 48, Issue 3, pp 515–532

Sufficient descent directions in unconstrained optimization


DOI: 10.1007/s10589-009-9268-z

Cite this article as:
An, X., Li, D. & Xiao, Y. Comput Optim Appl (2011) 48: 515. doi:10.1007/s10589-009-9268-z


Descent property is very important for an iterative method to be globally convergent. In this paper, we propose a way to construct sufficient descent directions for unconstrained optimization. We then apply the technique to derive a PSB (Powell-Symmetric-Broyden) based method. The PSB based method locally reduces to the standard PSB method with unit steplength. Under appropriate conditions, we show that the PSB based method with Armijo line search or Wolfe line search is globally and superlinearly convergent for uniformly convex problems. We also do some numerical experiments. The results show that the PSB based method is competitive with the standard BFGS method.


Unconstrained optimizationSufficient descent directionPSB methodGlobal convergenceSuperlinear convergence

Supplementary material

10589_2009_9268_MOESM1_ESM.pdf (387 kb)
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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.College of Mathematics and EconometricsHunan UniversityChangshaPeoples Republic of China
  2. 2.Institute of Applied Mathematics, College of Mathematics and Information ScienceHenan UniversityKaifengPeoples Republic of China