An efficient three-term conjugate gradient method with adaptive regularization and its application in image restoration

Conjugate gradient (CG) methods, valued for their low memory requirements, and quasi-Newton (QN) methods, known for fast convergence, are crucial for large-scale unconstrained optimization. Aiming to combine the strengths of these approaches, this paper proposes a new efficient three-term conjugate gradient method, designated as ATTCG-AR. This method modifies the framework of the accelerated NATTCG algorithm by introducing a novel adaptive regularization parameter. Formulated with inspiration from the philosophy of memoryless quasi-Newton methods (such as ADHCG), this parameter improves the three-term search direction by intelligently adjusting the gradient component. The resulting search direction ensures the sufficient descent property without reliance on the Powell restart strategy. The global convergence of the proposed algorithm is established under standard assumptions using the Wolfe line search conditions. Performance assessments via Dolan-Mor\'{e} profiles demonstrate the efficiency of the ATTCG-AR method on standard test problems, further validated by its application to salt-and-pepper noise removal in image restoration. Mathematics Subject Classifications (2000): 90C30, 90C53, 49M37, 65K05.