The modified Levenberg–Marquardt method with nonmonotone technique

Yuya Zheng; Yueting Yang; Mingyuan Cao; Yuya Zheng; Yueting Yang; Mingyuan Cao

doi:10.3934/math.2026102

AIMS Mathematics

2026, Volume 11, Issue 1: 2527-2546. doi: 10.3934/math.2026102

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Research article

The modified Levenberg–Marquardt method with nonmonotone technique

School of Mathematics and Statistics, Beihua University, Jilin 132013, China

Received: 17 October 2025 Revised: 07 January 2026 Accepted: 19 January 2026 Published: 26 January 2026
MSC : 65K05, 90C30

In this paper, we propose a modified Levenberg–Marquardt (LM) method with a nonmonotone technique for solving nonlinear equations. Under the $ \mathrm{H\ddot{o}lderian} $ continuity and the $ \mathrm{H\ddot{o}lderian} $ local error bounds conditions, which are weaker than the local error bounds and the Lipschitz continuity, the global convergence and local convergence of the algorithm are proved. Numerical experiments also show that the algorithm is effective.
- Levenberg–Marquardt method,
- nonlinear equations,
- convergence rate,
- nonmonotone technique,
- $ \mathrm{H\ddot{o}lderian} $ local error bounds
Citation: Yuya Zheng, Yueting Yang, Mingyuan Cao. The modified Levenberg–Marquardt method with nonmonotone technique[J]. AIMS Mathematics, 2026, 11(1): 2527-2546. doi: 10.3934/math.2026102

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Abstract

In this paper, we propose a modified Levenberg–Marquardt (LM) method with a nonmonotone technique for solving nonlinear equations. Under the $ \mathrm{H\ddot{o}lderian} $ continuity and the $ \mathrm{H\ddot{o}lderian} $ local error bounds conditions, which are weaker than the local error bounds and the Lipschitz continuity, the global convergence and local convergence of the algorithm are proved. Numerical experiments also show that the algorithm is effective.

References

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