A New Modified Three-Term Conjugate Gradient Method with Sufficient Descent Property and Its Global Convergence

Abstract

A new modified three-term conjugate gradient (CG) method is shown for solving the large scale optimization problems. The idea relates to the famous Polak-Ribière-Polyak (PRP) formula. As the numerator of PRP plays a vital role in numerical result and not having the jamming issue, PRP method is not globally convergent. So, for the new three-term CG method, the idea is to use the PRP numerator and combine it with any good CG formula's denominator that performs well. The new modification of three-term CG method possesses the sufficient descent condition independent of any line search. The novelty is that by using the Wolfe Powell line search the new modification possesses global convergence properties with convex and nonconvex functions. Numerical computation with the Wolfe Powell line search by using the standard test function of optimization shows the efficiency and robustness of the new modification.

A new smoothing modified three-term conjugate gradient method for l 1 $l_{1}$ -norm minimization problem

“…Using the techniques similar to lemmas in [ 31 – 33 ], we can get this lemma. The description will not be repeated again.…”

“…On the other hand, the nonlinear conjugate gradient methods and smoothing methods are used widely to solve large-scale optimization problems [ 24 , 25 ], total variation image restoration [ 26 ], monotone nonlinear equations with convex constraints [ 27 ], and nonsmooth optimization problems, such as nonsmooth nonconvex problems [ 28 ], minimax problem [ 29 ], P0 nonlinear complementarity problems [ 30 ]. Specially, the effectiveness of widely used and attained different numerical outcomes three-term conjugate gradient method, which is based on Hang–Zhang conjugate gradient method and Polak–Ribière–Polyak conjugate gradient method [ 31 – 33 ], has been widely studied. Based on the above analysis, in this paper, we propose a new smoothing modified three-term conjugate gradient method to solve problem ( 1 ).…”

A New Three-Term Conjugate Gradient-Based Projection Method for Solving Large-Scale Nonlinear Monotone Equations

“…Conjugate gradient methods [3,9,17,22] are very efficient methods for solving large-scale unconstrained optimization problems mainly due to their simplicity and low storage requirements. This has over the years influenced researchers to propose conjugate gradient-based projection methods by combining conjugate gradient methods with the hyperplane projection method [16] to solve nonlinear monotone equations.…”

A New Modified Three-Term Hestenes–Stiefel Conjugate Gradient Method with Sufficient Descent Property and Its Global Convergence

“…For instance, Beale [18] and Nazareth [19] proposed CG methods based on three terms that possess the finite termination property, but these do not perform well in practice [20,21]. Furthermore, reports by McGuire and Wolfe [22], Deng and Li [23], Zhang et al [24,25], Cheng [26], Al-Bayati and Sharif [27], Zhang Xiao and Wei [28], Andrei [29][30][31], Sugiki et al [32], Narushima et al [33], Babaie-Kafaki and Ghanbari [34], Al-Baali et al [35], Sun and Liu [36], and Baluch et al [37] discuss the global convergence and numerical results of modified three-term CG methods.…”