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This is a Quantitative Study study.

October 1, 2025Open Access

On the complexity of proximal gradient and proximal Newton-CG methods for \(_1\)-regularized Optimization

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Proximal gradient and Newton-CG methods improve iteration complexity in l1-regularized optimization, indicating enhanced convergence rates.

The proximal Newton-CG method shows the best iteration complexity, achieving a weak approximate second-order stationary point efficiently.
Numerical experiments confirm that both proposed methods can effectively navigate issues in finding first-order approximate solutions.
Comprehensive analyses cover iteration and computational complexity, highlighting local superlinear convergence under certain conditions.
Hybrid proximal gradient and negative curvature methods are designed to address unique approximate second-order stationary point definitions.

Zhu Hong (2025) studied this question.

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Proximal gradient and Newton-CG methods improve iteration complexity in l1-regularized optimization, indicating enhanced convergence rates.