Several Spectral Conjugate Gradient Methods

Spectral conjugate gradient method is a new iterative algorithm for solving large-scale unconstrained optimization.Its basic idea is to combine the spectral gradient method with the conjugate gradient method.Because the algorithm is simple,effective,and the storage is less,and the quadratic function is R-super linearly convergent,it has gradually attracted people’s attention and research.In this paper,based on previous research results,three kinds of spectral conjugate gradient methods are constructed.Firstly,the spectral coefficientδ_kis introduced on the basis of conjugate coefficientβ_k^RMIL+,which has been proposed in literature[37],and a new spectral conjugate pa-rameterβ_k^SNformula is obtained,thus constructing a new spectral conjugate gradient method.The search direction of the new method is sufficient descent property under any linear search rule.Under the standard Wolfe line search,the global convergence of the new algorithm is proved.Secondly,a new selection ofβ_kis given based on conjugate coefficientβ_k^RMIL+in the literature[37]and conjugate coefficientβ_k^*in the literature[32].The algorithm can automatically descend at every iteration.And this property depends neither on the line search used,nor on the convexity of the objective.Combined with Armijo linear search,we study the global convergence of the algorithm under general assumptions.Finally,inspired by the existing literature,we improve the spectral coefficientθ_k,and construct a new spectral conjugate gradient method based on the existing conjugate coefficientβ_k^RMILformula.The new algorithm satisfies the famous conjugacy condition:d_k^Ty_k-1=0,independent of any line search.The new method possesses the sufficient descent property without any line searches.Under the modified Wolfe linear search,the sufficient descent and global convergence of the algorithm are verified.