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<p>We propose two nonmonotone retraction-based proximal gradient methods for solving a class of nonconvex nonsmooth optimization problems over the Stiefel manifold. The proposed methods are equipped with the descent direction obtained by a proximal mapping restricted in tangent space of the manifold and the Barzilai-Borwein stepsizes determined by two recent iteration points and the corresponding descent directions. By employing, respectively, the Grippo-Lampariello-Lucidi nonmonotone line search strategy and the Dai-Fletcher nonmonotone line search strategy, our proposed methods are proved to be globally convergent. Analysis on the iteration complexity for obtaining an ϵ-stationary solution is provided. Numerical results on the sparse principle component analysis problems demonstrate the efficiency of our methods.</p>

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Two Nonmonotone Proximal Gradient Methods for Nonsmooth Optimization over the Stiefel Manifold

  • 期刊论文

  • 发表日期：

    2026年1月2日

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  • Jin-chao Zhang,
  • Juan Gao,
  • Ya-kui Huang,
  • Xin-wei Liu

摘要

We propose two nonmonotone retraction-based proximal gradient methods for solving a class of nonconvex nonsmooth optimization problems over the Stiefel manifold. The proposed methods are equipped with the descent direction obtained by a proximal mapping restricted in tangent space of the manifold and the Barzilai-Borwein stepsizes determined by two recent iteration points and the corresponding descent directions. By employing, respectively, the Grippo-Lampariello-Lucidi nonmonotone line search strategy and the Dai-Fletcher nonmonotone line search strategy, our proposed methods are proved to be globally convergent. Analysis on the iteration complexity for obtaining an ϵ-stationary solution is provided. Numerical results on the sparse principle component analysis problems demonstrate the efficiency of our methods.