A Family of Effective Spectral CG Methods With an Adaptive Restart Scheme

ABSTRACT In this paper, we propose a family of effective spectral conjugate gradient methods with an adaptive restart scheme for solving unconstrained optimization problems. First, we construct a new composite conjugate parameter with two parameters by employing a convex combination of classical conjugate parameters and their variants. Then, we use the spectral technique to guarantee that the search direction possesses the sufficient descent property independent of any line search. Additionally, we incorporate a new spectral gradient‐based adaptive restart scheme to ensure the global convergence of the family under weak Wolfe line search and obtain iteration complexity results under Armijo line search. Finally, we conduct numerical experiments on unconstrained optimization problems and image restoration applications to demonstrate the effectiveness and practicality of the proposed family.