A class of spectral three-term descent Hestenes-Stiefel conjugate gradient algorithms for large-scale unconstrained optimization and image restoration problems.

Conjugate gradient methods are much effective and widely used for large-scale unconstrained optimization problems by their simple computation, low memory requirement and strong global convergence property. Spectral gradient methods are also effective for large-scale problems. In this paper, a class of new descent spectral three-term conjugate gradient algorithms are proposed which automatically have the sufficient descent property and satisfy the Dai-Liao conjugate condition. Under the Wolfe line search technique and some standard conditions, the proposed methods are globally convergent for strongly convex functions and general nonlinear functions with the help of the modified secant equations. In numerical part, 732 problems with dimensions varying from 1500 to 150000 and three image restoration problems with three noise levels are considered. Numerical results indicate that the proposed algorithms are more efficient, reliable and robust than the other methods for the testing problems. • A class of spectral descent three-term conjugate gradient methods are proposed. • The proposed methods own the sufficient descent property and satisfy the Dai-Liao conjugate condition automatically. • The proposed methods are globally convergent for general functions without convex assumptions. • The proposed methods are applied to large-scale unconstrained optimization problems and some image restoration problems. [ABSTRACT FROM AUTHOR]