Spectral Conjugate Gradient Method And Three-term Conjugate Gradient Method With Applications In Image Restoration

The conjugate gradient method(CGM)is very welcome to solve large-scale unconstrained optimization problems because of its simplicity of the iterations and its lower storage requirements.In this paper,we focus on the spectral conjugate gradient method(SCGM)and hybrid three-term conjugate gradient method(HTTCGM)and their applications in image restoration,which mainly content included in two parts.The first work of this paper,the Polak-Ribière-Polak algorithm is considered one of the most efficient methods among classical CGMs.To generate new conjugate parameter,an improved PRP formula is proposed by combining the strong Wolfe line search condition.Furthermore,a new spectral parameter and a new restart direction are designed,and thus a new SCGM with restart steps is established.Using the strong Wolfe line search condition to yield the step length,the sufficient descent property and strong convergence of the new algorithm are obtained under the general assumptions.Then,for the proposed algorithm,a mediumlarge scale numerical experiments is performed,and compared with some existing efficient CGMs.The search direction of the new algorithm is applied to solve nonlinear monotone equations and then image restoration.Experimental results show that the proposed algorithm is very effective in both unconstrained optimization problems and image restoration.The second work of this paper,based on the hybrid conjugate gradient method and the convex combination technique,a new family of HTTCGMs are proposed for solving unconstrained optimization.The conjugate parameter in the search direction is a hybrid of Dai-Yuan conjugate parameter and any one.The search direction then is the sum of the negative gradient direction and a convex combination in relation to the last search direction and the gradient at the previous iteration.Without choosing any specific conjugate parameters,we show that the search direction generated by the family always possesses the descent property independent of line search technique,and that it is globally convergent under usual assumptions and the weak Wolfe line search.To verify the effectiveness of the presented family,we further design a specific conjugate parameter,and perform medium-large-scale numerical experiments for smooth unconstrained optimization and image restoration problems.Experimental results show that the proposed methods are very effective.