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

A Three-Term Conjugate Gradient Algorithm with Quadratic Convergence for Unconstrained Optimization Problems

“…and we all know that is sufficiently descent by restricting the parameter 2 < 1/4 under the strong Wolfe-Powell linear search [33]. By the definition of in (4), for ≥ 1, we get…”

“…It is easy to deduce that = ( − (‖ ‖/‖ −1 ‖) −1 )/‖ −1 ‖ 2 ≥ (‖ ‖ − (‖ ‖/ ‖ −1 ‖)‖ −1 ‖)/‖ −1 ‖ 2 = 0. The global convergences of the WYL algorithm with exact linear search, the Grippo-Lucidi Armijo line search, and the Wolfe-Powell line search have been established by [21,[33][34][35]. By restricting the parameter 2…”

A modified conjugate gradient algorithm for optimization problems

“…Under assumption of the sufficient descent condition, the new method with the WWP line search is globally convergent. Huang et al [18] proved that, by restricting the parameter σ < 1 4 , under the strong WolfePowell (SWP) line search rule, this method had the sufficient descent property. However, the sufficient descent property of this method with WWP line search can not be satisfied.…”

A hybrid conjugate gradient method with descent property for unconstrained optimization

“…Furthermore, Huang et al [11] proved that the WYL method satisfies the sufficient descent condition and converges globally under the strong Wolfe line search (5) if the parameter satisfies r < 1 4 . Yao et al [12] extended this idea to the HS method and proposed a conjugate gradient method, denote it by YWH method, that is…”

“…In the latest years, based on the above four formulas and their hybridization, many works putting effort into seeking for new CGMs with not only good convergence property but also excellent numerical effect were published, see Refs. [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26].…”