The proof of the sufficient descent condition of the Wei–Yao–Liu conjugate gradient method under the strong Wolfe–Powell line search

In [Z. Wei et al., The convergence properties of some conjugate gradient methods, Appl. Math. Comput. 183 (2006) 1341–1350], Wei et al. proposed a new conjugate gradient method called WYL method which has good numerical experiments. Under the sufficient descent condition and WWP line search, the given method’s convergent results have been established by Wei et al. So, the sufficient descent condition is important for the WYL method. In this paper, we prove that, by restricting the parameter σ<14, under the SWP line search condition, the WYL method possess the sufficient descent condition which deduces the global convergent results of the WYL method under SWP line search.