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<p>Gradient method is an important method for solving large scale problems. In this paper, a new gradient method framework for unconstrained optimization problem is proposed, where the stepsize is updated in a cyclic way. The Cauchy step is approximated by the quadratic interpolation. And the cycle for stepsize update is adjusted adaptively. Combining with the adaptive nonmonotone line search technique, we prove the global convergence of the proposed method. Furthermore, its sublinear convergence rate for convex problems and <i>R</i>-linear convergence rate for problems with quadratic functional growth property are analyzed. Numerical results show that our proposed algorithm enjoys good performances in terms of both computational cost and obtained function values.</p>

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Adaptive cyclic gradient methods with interpolation

  • 期刊论文

  • 发表日期：

    2025年5月13日

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  • Yixin Xie,
  • Cong Sun,
  • Ya-Xiang Yuan

摘要

Gradient method is an important method for solving large scale problems. In this paper, a new gradient method framework for unconstrained optimization problem is proposed, where the stepsize is updated in a cyclic way. The Cauchy step is approximated by the quadratic interpolation. And the cycle for stepsize update is adjusted adaptively. Combining with the adaptive nonmonotone line search technique, we prove the global convergence of the proposed method. Furthermore, its sublinear convergence rate for convex problems and R-linear convergence rate for problems with quadratic functional growth property are analyzed. Numerical results show that our proposed algorithm enjoys good performances in terms of both computational cost and obtained function values.