Two families of hybrid conjugate gradient methods with restart procedures and their applications

In this paper, two families of hybrid conjugate gradient methods with restart procedures are proposed. Their hybrid conjugate parameters are yielded by projection or convex combination of the classical parameters. Moreover, their restart procedures are given uniformly, which are determined by the proposed hybrid conjugate parameters. The search directions of the presented families satisfy the sufficient descent condition. Under usual assumption and the weak Wolfe line search, the proposed families are proved to be globally convergent. Finally, choosing a specific parameter for each family to solve large-scale unconstrained optimization problems, convex constrained nonlinear monotone equations and image restoration problems. All the numerical results are reported and analysed, which show that the proposed families of hybrid conjugate gradient methods are promising.