A generalized self-regular Kernel function for large-scale nonlinear optimization problems

Mounia Laouar; Mahmoud Brahimi; Raouf Ziadi; Mohammed A. Saleh; Abdulgader Z. Almaymuni; Benmessaoud Chahinez; Mounia Laouar; Mahmoud Brahimi; Raouf Ziadi; Mohammed A. Saleh; Abdulgader Z. Almaymuni; Benmessaoud Chahinez

doi:10.3934/math.2026202

AIMS Mathematics

2026, Volume 11, Issue 2: 4935-4965. doi: 10.3934/math.2026202

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A generalized self-regular Kernel function for large-scale nonlinear optimization problems

1.
Laboratory of Partial Differential Equations, Department of Mathematics, Faculty of Mathematics and Computer Science, University of Batna 2, Batna, Algeria
2.
Laboratory of Fundamental and Numerical Mathematics (LMFN), Department of Mathematics, University Setif -1- Ferhat Abbas, Setif, Algeria
3.
Department of Cybersecurity, College of Computer, Qassim University, Saudi Arabia
4.
University of Batna 2, Batna 05000, Algeria

Received: 09 January 2026 Revised: 06 February 2026 Accepted: 11 February 2026 Published: 27 February 2026
MSC : 90C20, 90C25, 90C30, 90C51

This work investigated the computational efficiency of primal-dual interior-point methods for nonlinear convex optimization by refining both the underlying kernel functions and the barrier parameter update mechanisms. We introduced a unified parametric class of self-regular kernels that generalizes several established barrier families while maintaining optimal theoretical iteration complexity. To bridge the gap between theoretical convergence and practical performance, we proposed an adaptive update rule for the barrier parameter and evaluated various heuristics for its dynamic selection. Extensive numerical testing on a diverse benchmark suite demonstrated that the proposed framework significantly outperforms the Interior Point OPTimizer (IPOPT) solver while maintaining high numerical accuracy and minimal stationarity residuals. Moreover, the framework exhibited robust performance even on nonconvex problems, highlighting its practical versatility beyond the theoretical convex setting.
- optimization method,
- nonlinear convex optimization,
- interior-point methods,
- self-regular kernels,
- large-scale problems
Citation: Mounia Laouar, Mahmoud Brahimi, Raouf Ziadi, Mohammed A. Saleh, Abdulgader Z. Almaymuni, Benmessaoud Chahinez. A generalized self-regular Kernel function for large-scale nonlinear optimization problems[J]. AIMS Mathematics, 2026, 11(2): 4935-4965. doi: 10.3934/math.2026202

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Abstract

This work investigated the computational efficiency of primal-dual interior-point methods for nonlinear convex optimization by refining both the underlying kernel functions and the barrier parameter update mechanisms. We introduced a unified parametric class of self-regular kernels that generalizes several established barrier families while maintaining optimal theoretical iteration complexity. To bridge the gap between theoretical convergence and practical performance, we proposed an adaptive update rule for the barrier parameter and evaluated various heuristics for its dynamic selection. Extensive numerical testing on a diverse benchmark suite demonstrated that the proposed framework significantly outperforms the Interior Point OPTimizer (IPOPT) solver while maintaining high numerical accuracy and minimal stationarity residuals. Moreover, the framework exhibited robust performance even on nonconvex problems, highlighting its practical versatility beyond the theoretical convex setting.

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