Modified fletcher–reeves spectral conjugate gradient algorithms with three-term structure for optimization and image restoration

The conjugate gradient (CG) method is a widely used technique for solving large-scale unconstrained optimization problems due to its simplicity and low memory requirements. The classical Fletcher-Reeves (FR) method, known for its strong theoretical convergence properties, often exhibits slow practical performance and is prone to jamming when used with inexact line searches or on poorly scaled problems. To overcome these limitations, we propose two new spectral three-term conjugate gradient algorithms, TTUHS1 and TTUHS2, which incorporate spectral scaling and a three-term update structure inspired by FR. The proposed algorithms are designed to retain the global convergence and descent properties of FR while significantly improving numerical efficiency. Extensive numerical experiments on standard test functions and image restoration problems demonstrate that TTUHS1 and TTUHS2 outperform the Three-Term Fletcher-Reeves (TTFR) algorithm in terms of convergence speed, iteration count, and robustness, particularly in large-scale problems.