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Generalized damped Newton algorithms in nonsmooth optimization via second-order subdifferentials

Author

Listed:
  • Pham Duy Khanh

    (HCMC University of Education)

  • Boris S. Mordukhovich

    (Wayne State University)

  • Vo Thanh Phat

    (HCMC University of Education
    Wayne State University)

  • Dat Ba Tran

    (Wayne State University)

Abstract

The paper proposes and develops new globally convergent algorithms of the generalized damped Newton type for solving important classes of nonsmooth optimization problems. These algorithms are based on the theory and calculations of second-order subdifferentials of nonsmooth functions with employing the machinery of second-order variational analysis and generalized differentiation. First we develop a globally superlinearly convergent damped Newton-type algorithm for the class of continuously differentiable functions with Lipschitzian gradients, which are nonsmooth of second order. Then we design such a globally convergent algorithm to solve a structured class of nonsmooth quadratic composite problems with extended-real-valued cost functions, which typically arise in machine learning and statistics. Finally, we present the results of numerical experiments and compare the performance of our main algorithm applied to an important class of Lasso problems with those achieved by other first-order and second-order optimization algorithms.

Suggested Citation

  • Pham Duy Khanh & Boris S. Mordukhovich & Vo Thanh Phat & Dat Ba Tran, 2023. "Generalized damped Newton algorithms in nonsmooth optimization via second-order subdifferentials," Journal of Global Optimization, Springer, vol. 86(1), pages 93-122, May.
    • Handle: RePEc:spr:jglopt:v:86:y:2023:i:1:d:10.1007_s10898-022-01248-7
    DOI: 10.1007/s10898-022-01248-7
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    References listed on IDEAS

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    1. Nguyen T. V. Hang & Boris S. Mordukhovich & M. Ebrahim Sarabi, 2022. "Augmented Lagrangian method for second-order cone programs under second-order sufficiency," Journal of Global Optimization, Springer, vol. 82(1), pages 51-81, January.
    2. Yurii Nesterov, 2018. "Lectures on Convex Optimization," Springer Optimization and Its Applications, Springer, edition 2, number 978-3-319-91578-4, December.
    3. Robert Tibshirani & Michael Saunders & Saharon Rosset & Ji Zhu & Keith Knight, 2005. "Sparsity and smoothness via the fused lasso," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(1), pages 91-108, February.
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    5. Hui Zou & Trevor Hastie, 2005. "Addendum: Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(5), pages 768-768, November.
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