A Chebyshev–Halley Method with Gradient Regularization and an Improved Convergence Rate

Jianyu Xiao, Haibin Zhang and Huan Gao ()
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Jianyu Xiao: Beijing Institute for Scientific and Engineering Computing, Beijing University of Technology, Beijing 100124, China
Haibin Zhang: Beijing Institute for Scientific and Engineering Computing, Beijing University of Technology, Beijing 100124, China
Huan Gao: College of Mathematics and Computational Science, Hunan First Normal University, Changsha 410205, China

Mathematics, 2025, vol. 13, issue 8, 1-17

Abstract: High-order methods are particularly crucial for achieving highly accurate solutions or satisfying high-order optimality conditions. However, most existing high-order methods require solving complex high-order Taylor polynomial models, which pose significant computational challenges. In this paper, we propose a Chebyshev–Halley method with gradient regularization, which retains the convergence advantages of high-order methods while effectively addressing computational challenges in polynomial model solving. The proposed method incorporates a quadratic regularization term with an adaptive parameter proportional to a certain power of the gradient norm, thereby ensuring a closed-form solution at each iteration. In theory, the method achieves a global convergence rate of O ( k − 3 ) or even O ( k − 5 ) , attaining the optimal rate of third-order methods without requiring additional acceleration techniques. Moreover, it maintains local superlinear convergence for strongly convex functions. Numerical experiments demonstrate that the proposed method compares favorably with similar methods in terms of efficiency and applicability.

Keywords: Chebyshev–Halley method; quadratic regularization; gradient norm; closed-form solution; global convergence; local superlinear convergence (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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