Alternative structured spectral gradient algorithms for solving nonlinear least-squares problems

Mahmoud Muhammad Yahaya; Poom Kumam; Aliyu Muhammed Awwal; Sani Aji

doi:10.1016/j.heliyon.2021.e07499

Alternative structured spectral gradient algorithms for solving nonlinear least-squares problems

Heliyon. 2021 Jul 7;7(7):e07499. doi: 10.1016/j.heliyon.2021.e07499. eCollection 2021 Jul.

Authors

Mahmoud Muhammad Yahaya^{1

2}, Poom Kumam^{1

2}, Aliyu Muhammed Awwal^{1

3

4}, Sani Aji^{1

4}

Affiliations

¹ Center of Excellence in Theoretical and Computational Science (TaCS-CoE) and KMUTTFixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory Science Laboratory Building, Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand.
² Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi, 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand.
³ NCAO Research Center, Fixed Point Theory and Applications Research Group, Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand.
⁴ Department of Mathematics, Faculty of Science, Gombe State University, Gombe 760214, Nigeria.

Abstract

The study of efficient iterative algorithms for addressing nonlinear least-squares (NLS) problems is of great importance. The NLS problems, which belong to a special class of unconstrained optimization problems, are of particular interest because of the special structure of their gradients and Hessians. In this paper, based on the spectral parameters of Barzillai and Borwein (1998), we propose three structured spectral gradient algorithms for solving NLS problems. Each spectral parameter in the respective algorithms incorporates the structured gradient and the information gained from the structured Hessian approximation. Moreover, we develop a safeguarding technique for the first two structured spectral parameters to avoid negative curvature directions. Moreso, using a nonmonotone line-search strategy, we show that the proposed algorithms are globally convergent under some standard conditions. The comparative computational results on some standard test problems show that the proposed algorithms are efficient.

Keywords: Iterative algorithm; Line–search; Nonlinear least squares; Quasi–Newton algorithm; Spectral gradient algorithm.