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

Jorge Nocedal, Annick Sartenaer, Ciyou Zhu

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    Abstract

    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
    Pages (from-to)5-35
    Number of pages31
    JournalComputational Optimization and Application
    Volume22
    Issue number1
    Publication statusPublished - 2002

    Keywords

    • nonlinear optimization
    • unconstrained optimization
    • behavior of the gradeint norm
    • steepest descent method

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