On the behavior of the gradient norm in the steepest descent method

Jorge Nocedal*, Annick Sartenaer, Ciyou Zhu

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

37 Scopus citations


It is well known that the norm of the gradient may be unreliable as a stopping test in unconstrained optimization, and that it often exhibits oscillations in the course of the optimization. In this paper we present results describing the properties of the gradient norm for the steepest descent method applied to quadratic objective functions. We also make some general observations that apply to nonlinear problems, relating the gradient norm, the objective function value, and the path generated by the iterates.

Original languageEnglish (US)
Pages (from-to)5-35
Number of pages31
JournalComputational Optimization and Applications
Issue number1
StatePublished - Apr 2002


  • Behavior of gradient norm
  • Nonlinear optimization
  • Steepest descent method
  • Unconstrained optimization

ASJC Scopus subject areas

  • Control and Optimization
  • Computational Mathematics
  • Applied Mathematics

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