*Article* **A Spectral Conjugate Gradient Method with Descent Property**

#### **Jinbao Jian 1, Lin Yang 1, Xianzhen Jiang 1,\*, Pengjie Liu 2 and Meixing Liu 3**


Received: 6 January 2020; Accepted: 13 February 2020; Published: 19 February 2020

**Abstract:** Spectral conjugate gradient method (SCGM) is an important generalization of the conjugate gradient method (CGM), and it is also one of the effective numerical methods for large-scale unconstrained optimization. The designing for the spectral parameter and the conjugate parameter in SCGM is a core work. And the aim of this paper is to propose a new and effective alternative method for these two parameters. First, motivated by the strong Wolfe line search requirement, we design a new spectral parameter. Second, we propose a hybrid conjugate parameter. Such a way for yielding the two parameters can ensure that the search directions always possess descent property without depending on any line search rule. As a result, a new SCGM with the standard Wolfe line search is proposed. Under usual assumptions, the global convergence of the proposed SCGM is proved. Finally, by testing 108 test instances from 2 to 1,000,000 dimensions in the CUTE library and other classic test collections, a large number of numerical experiments, comparing with both SCGMs and CGMs, for the presented SCGM are executed. The detail results and their corresponding performance profiles are reported, which show that the proposed SCGM is effective and promising.

**Keywords:** unconstrained optimization; spectral conjugate gradient method; Wolfe line search; global convergence
