Conjugate gradient techniques: Enhancing optimization efficiency for large-scale problems and image restoration

Conjugate gradient methods are widely recognized for their efficiency in solving large-scale optimization problems with minimal memory requirements. This paper introduces a new CG algorithm derived from a modified secant condition, utilizing a redefined difference gradient vector to improve numerical stability and optimization performance. By integrating this modified vector into the computation of the conjugacy parameter $ \beta_k $, we propose a new parmeter, $ \beta_k^{ABM} $, which effectively resolves the issues of non-negativity and convergence that are encountered in existing methods. Numerical experiments confirm the superior performance of the proposed method on unconstrained optimization problems across various dimensions, demonstrating faster convergence rates, fewer iterations, and reduced computational times compared to traditional CG (HS, DL, and LS) methods. Additionally, the algorithm's practical effectiveness is validated in medical image restoration tasks, specifically in reducing salt-and-pepper noise in grayscale and color images. When evaluated against established methods like HS, FHTTCGM-N, and HZ, the proposed algorithm achieves notable improvements in metrics such as iteration count, computational time, mean squared error, peak signal-to-noise ratio, and structural similarity index. These results underscore the algorithms significant contributions to both the theoretical and practical advancements of CG methods in optimization and image restoration.