Computational efficiency and global convergence of modified RMIL conjugate gradient methods for unconstrained optimization

Purpose This paper aims to develop a new family of self-scaling conjugate gradient (CG) methods for efficiently solving large-scale unconstrained optimization problems and to demonstrate their effectiveness in practical applications such as image restoration. Design/methodology/approach The proposed algorithms incorporate two optimal scaling parameters: one obtained by minimizing a measure function associated with the search direction matrix, and another derived by mimicking the Newton direction. These strategies ensure that the resulting search directions satisfy a sufficient descent condition independently of the line search accuracy, while preserving a matrix-free structure suitable for large-scale problems. Global convergence is established under standard assumptions. The methods are evaluated using extensive numerical experiments on CUTEr benchmark problems and are further applied to grayscale image restoration corrupted by salt-and-pepper noise. Findings Numerical results show that the proposed methods outperform several state-of-the-art CG algorithms in terms of iteration count, computational efficiency and robustness on large-scale CUTEr test problems. In image restoration experiments, the methods achieve faster convergence and higher peak signal-to-noise ratio (PSNR) compared with existing approaches, particularly in the presence of high-level impulse noise. Originality/value This study introduces a novel family of self-scaling CG methods that combine optimal scaling strategies with Newton-inspired directions to guarantee sufficient descent without relying on highly accurate line searches. The integration of these techniques yields efficient, robust and matrix-free algorithms, providing both theoretical and practical advances in large-scale optimization and image restoration.