An inexact conjugate gradient method for symmetric nonlinear equations

Abstract

In this article, we present a conjugate gradient method for large‐scale nonlinear equations with symmetric Jacobian. The method is a modification of the descent Dai‐Liao conjugate gradient method for unconstrained optimization problem proposed by Kafaki and Gambari. Under some assumptions, which include symmetric property of the Jacobian, we establish the global convergence of the proposed method and, finally, some numerical results to show its efficiency.

A Modified PRP-CG Type Derivative-Free Algorithm with Optimal Choices for Solving Large-Scale Nonlinear Symmetric Equations

“…In this section, we compare the numerical performances of the modified PRP CGtype algorithm using the proposed optimal choice with the norm descent derivative-free algorithm (NDDA) [21] and the ICGM algorithm [22] for solving the nonlinear symmetric Equation (1). For the MPRP algorithm, we set: ζ = 10 −4 , a = 0.4, t 0 = 0.01 and φ k = 1 (10 4 +k) 2 .…”

“…Similarly, the proposed algorithm remained the most stable algorithms concerning the number of CPU time and the number of function evaluations as their curves correspond to the top left curves. [21] and ICGM [22] for number of function evaluations.…”

“…Besides, Zhou and Shen have suggested an inexact version of the PRP CG method for the systems of symmetric nonlinear equations [14]. Since then, different CG methods for the solution of symmetric nonlinear systems have been presented, see [15][16][17][18][19][20][21][22][23] for more details. Moreover, motivated by the efficiency of the extended PRP method [11] and the robustness of CG methods in solving large-scale symmetric nonlinear systems, we want to propose two optimal relations for the computation of η in (8) and use the corresponding modified PRP parameters to propose algorithms for solving large-scale symmetric nonlinear systems without using the exact gradient information.…”

Least-Square-Based Three-Term Conjugate Gradient Projection Method for ℓ1-Norm Problems with Application to Compressed Sensing

“…We note that the sequences {t k }, {u k }, {F(t k )}, and {F(u k )} are bounded from (25), (31), (27), and (28), respectively. In addition, from the Lipchitz continuity and (25), we have…”

“…The method was also successfully used to solve the sparse signal in compressive sensing. Interested readers may refer to the following articles [21][22][23][24][25][26][27] for an overview of algorithms used for solving monotone operator equations.…”