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

Mahmoud Muhammad Yahaya1,2, Poom Kumam1,2, Aliyu Muhammed Awwal1,3,4

  • 1Center 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.

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This study introduces three novel spectral gradient algorithms for nonlinear least-squares problems. These methods enhance efficiency and global convergence for optimization tasks.

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Area of Science:

  • Numerical Analysis
  • Optimization Theory

Background:

  • Nonlinear least-squares (NLS) problems are a significant class of unconstrained optimization problems.
  • The special structure of gradients and Hessians in NLS problems warrants specialized algorithms.

Purpose of the Study:

  • To propose three structured spectral gradient algorithms for solving NLS problems.
  • To enhance the efficiency and global convergence of iterative methods for NLS.

Main Methods:

  • Algorithms based on Barzilai and Borwein spectral parameters (1998).
  • Incorporation of structured gradients and Hessian approximations into spectral parameters.
  • Development of a safeguarding technique to prevent negative curvature directions.
  • Application of a nonmonotone line-search strategy.

Main Results:

  • The proposed algorithms effectively solve NLS problems.
  • Demonstrated global convergence under standard conditions.
  • Comparative computational results indicate high efficiency on test problems.

Conclusions:

  • The developed structured spectral gradient algorithms are efficient for NLS problems.
  • The safeguarding technique and nonmonotone line-search contribute to robust convergence.
  • These algorithms represent an advancement in solving NLS optimization problems.