Global convergence of a modified spectral three-term CG algorithm for nonconvex unconstrained optimization problems

Spectral conjugate gradient methods are an efficient family for solving unconstrained optimization problems that have been widely studied in recent decades. In this regard, Li et al. (2019) proposed a spectral three-term conjugate gradient method and proved the global convergence of this algorithm for uniformly convex functions. Our main motivation in this paper is to develop the convergence properties of this method such that the new method possesses suitable convergence for general nonlinear functions. To do end, we introduce a modified spectral conjugate gradient method based on the CG method by Li et al. (2019). We show that the new method fulfills the sufficient descent property without any line search. The new algorithm is globally convergent for general nonlinear functions without the convexity assumption on the objective function. The numerical results indicate that the behavior of the new algorithm is not only effective, but also promising versus other conjugate gradient methods dealing with unconstrained optimization problems of the CUTEst library.