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A Semismooth Newton-Based Augmented Lagrangian Algorithm for the Generalized Convex Nearly Isotonic Regression Problem

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  • Yanmei Xu

    (School of Mathematics and Statistics, Fuzhou University, Fuzhou 350108, China)

  • Lanyu Lin

    (School of Mathematics and Statistics, Fuzhou University, Fuzhou 350108, China)

  • Yong-Jin Liu

    (Center for Applied Mathematics of Fujian Province, School of Mathematics and Statistics, Fuzhou University, Fuzhou 350108, China)

Abstract

The generalized convex nearly isotonic regression problem addresses a least squares regression model that incorporates both sparsity and monotonicity constraints on the regression coefficients. In this paper, we introduce an efficient semismooth Newton-based augmented Lagrangian ( Ssnal ) algorithm to solve this problem. We demonstrate that, under reasonable assumptions, the Ssnal algorithm achieves global convergence and exhibits a linear convergence rate. Computationally, we derive the generalized Jacobian matrix associated with the proximal mapping of the generalized convex nearly isotonic regression regularizer and leverage the second-order sparsity when applying the semismooth Newton method to the subproblems in the Ssnal algorithm. Numerical experiments conducted on both synthetic and real datasets clearly demonstrate that our algorithm significantly outperforms first-order methods in terms of efficiency and robustness.

Suggested Citation

  • Yanmei Xu & Lanyu Lin & Yong-Jin Liu, 2025. "A Semismooth Newton-Based Augmented Lagrangian Algorithm for the Generalized Convex Nearly Isotonic Regression Problem," Mathematics, MDPI, vol. 13(3), pages 1-18, February.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:3:p:501-:d:1582385
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    References listed on IDEAS

    as
    1. J. Han & D. Sun, 1997. "Newton and Quasi-Newton Methods for Normal Maps with Polyhedral Sets," Journal of Optimization Theory and Applications, Springer, vol. 94(3), pages 659-676, September.
    2. Yong-Jin Liu & Jing Yu, 2022. "A Semismooth Newton-based Augmented Lagrangian Algorithm for Density Matrix Least Squares Problems," Journal of Optimization Theory and Applications, Springer, vol. 195(3), pages 749-779, December.
    3. Defeng Sun & Jie Sun, 2002. "Semismooth Matrix-Valued Functions," Mathematics of Operations Research, INFORMS, vol. 27(1), pages 150-169, February.
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