A computationally improved Levenberg−Marquardt method for systems of inequalities

Kaile Ren; Jingyong Tang; Kaile Ren; Jingyong Tang

doi:10.3934/jimo.2026116

Journal of Industrial and Management Optimization

2026, Volume 22, Issue 7: 3175-3187. doi: 10.3934/jimo.2026116

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A computationally improved Levenberg−Marquardt method for systems of inequalities

Kaile Ren ,
Jingyong Tang ^,

School of Mathematics and Statistics, Henan Provincial Center for Applied Mathematics, Xinyang Normal University, Xinyang 464000, China

Received: 31 October 2025 Revised: 09 March 2026 Accepted: 21 May 2026 Published: 11 June 2026
90C33, 65K05

This paper addresses a system of inequalities by first transforming it into a system of nonlinear equations. We then develop a computationally enhanced Levenberg−Marquardt (LM) method to solve this equivalent system. In contrast to the standard LM approach, which solves a large linear system, the proposed method requires solving a linear system of significantly reduced dimensions at each iteration, leading to considerable computational savings. Global convergence is ensured through the incorporation of a derivative-free line search. We demonstrate that any accumulation point of the generated sequence is a solution to the original inequality system, and that a superlinear convergence rate is achievable under a local error bound condition. A key advantage of this method is its consistent ability to compute exact solutions to the inequality system. Numerical experiments validate the effectiveness of the proposed LM method and highlight its superior performance over the standard LM method, particularly for large-scale problems.
- system of inequalities,
- Levenberg−Marquardt method,
- superlinear convergence
Citation: Kaile Ren, Jingyong Tang. A computationally improved Levenberg−Marquardt method for systems of inequalities[J]. Journal of Industrial and Management Optimization, 2026, 22(7): 3175-3187. doi: 10.3934/jimo.2026116

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Abstract

This paper addresses a system of inequalities by first transforming it into a system of nonlinear equations. We then develop a computationally enhanced Levenberg−Marquardt (LM) method to solve this equivalent system. In contrast to the standard LM approach, which solves a large linear system, the proposed method requires solving a linear system of significantly reduced dimensions at each iteration, leading to considerable computational savings. Global convergence is ensured through the incorporation of a derivative-free line search. We demonstrate that any accumulation point of the generated sequence is a solution to the original inequality system, and that a superlinear convergence rate is achievable under a local error bound condition. A key advantage of this method is its consistent ability to compute exact solutions to the inequality system. Numerical experiments validate the effectiveness of the proposed LM method and highlight its superior performance over the standard LM method, particularly for large-scale problems.

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