The global convergence of a modified BFGS method for nonconvex functions

A conjugate gradient algorithm and its application in large-scale optimization problems and image restoration

“…where ϕ ∈ (0, 1/2), ρ ∈ (ϕ, 1) and ϕ 1 ∈ (0, ϕ). Further study can be found in [45]. Based on the innovation of the YWL line search technique, Yuan et al [41] focused on the usual Armijo line search technique and proposed a modified Armijo line search technique as follows: 9) where λ, γ ∈ (0, 1), λ 1 ∈ (0, λ), and α k is the largest number of {γ k | k = 0, 1, 2, .…”

The global proof of the Polak–Ribière–Polak algorithm under the YWL inexact line search technique

“…Here we call it YWL line search. It is used for not only the PRP method but also the BFGS quasi-Newton method (see [26,28] in detail). In the case min[-δ 1 g(x k ) T d k , δα k d k 2 ] = δα k d k 2 , the global convergence of the PRP algorithm is established including the conditions d T k g k < 0 and g k+1 d k ≤ -σ 1 g T k d k .…”

A Conjugate Gradient Algorithm under Yuan-Wei-Lu Line Search Technique for Large-Scale Minimization Optimization Models

“…Yuan et al [9] partly solved these two open problems under the YWL technique, and the numerical performance shows that the YWL technique is more competitive than the normal weak Wolfe-Powell technique. Further study work can be found in their paper [11]. By (5), it is not difficult to see that the YWL conditions are equivalent to the weak Wolfe-Powell (WWP) conditions if − 1 < ( /2)‖ ‖ 2 holds, which implies that the YWL technique includes the WWP technique in some sense.…”