Conjugate Gradient Methods With Restart Procedures And Their Applications In Image Restoration

In this paper,the conjugate gradient method for smooth unconstrained optimizations and systems of convex constrained equations with restart procedures is researched to deal with the image restoration problem.The specific content mainly includes three parts.The first work of this paper,two families of hybrid conjugate gradient method with restart procedures are proposed for the unconstrained optimization problems.Their hybrid conjugate parameters are yielded by projection or convex combination of the classical parameters.Moreover,the search direction sets the restart condition and restart direction related to the hybrid conjugation parameter,and satisfies 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 medium-large-scale unconstrained optimization and image restoration problems.All the numerical results are reported and analyzed,which show that the proposed families of hybrid conjugate gradient methods are promising.The second work of this paper,based on the Liu-Storey conjugate gradient method,a three-term conjugate gradient method with restart procedure is proposed for the unconstrained optimization problems.The designed restart condition is chosen according to the Liu-Storey conjugate parameter,and the restart direction consists of the first and third items in the nonrestart direction,which can be picked flexibly.Without depending on any line search,the direction in the new algorithm satisfies the sufficient descent condition.Moreover,the proposed algorithm possesses the global convergence under usual assumptions and the weak Wolfe line search.Finally,medium-large-scale numerical experiments are conducted for solving unconstrained optimization and image restoration problems,and compared with several methods which are recognized to have good numerical results.The numerical results show that the proposed algorithm is effective for solving the two kinds of problems.The third work of this paper,based on the idea of search direction design in the second part,an improved three-term PRP-type search direction with restart procedure is proposed,and a modified adaptive line search criterion is also given for systems of convex constrained equations.Thus,a three-term conjugate gradient method for systems of constrained equations with restart procedure is established.Without depending on any line search,the direction generated by the new algorithm satisfies the sufficient descent condition and the trust region property.Without considering Lipschitz continuity,the proposed algorithm is proved to be globally convergent.Furthermore,its R-linear convergence rate is reached under Lipschitz continuity and the usual assumptions.Finally,medium-large-scale numerical experiments for systems of convex constrained equations and image restorations have been performed,which show that the proposed algorithm is effective.