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

Abstract

Inspired by the large number of applications for symmetric nonlinear equations, this article will suggest two optimal choices for the modified Polak–Ribiére–Polyak (PRP) conjugate gradient (CG) method by minimizing the measure function of the search direction matrix and combining the proposed direction with the default Newton direction. In addition, the corresponding PRP parameters are incorporated with the Li and Fukushima approximate gradient to propose two robust CG-type algorithms for finding solutions for large-scale systems of symmetric nonlinear equations. We have also demonstrated the global convergence of the suggested algorithms using some classical assumptions. Finally, we demonstrated the numerical advantages of the proposed algorithms compared to some of the existing methods for nonlinear symmetric equations.

A Sparse Quasi-Newton Method Based on Automatic Differentiation for Solving Unconstrained Optimization Problems

“…Tables 2 and 3 contain the numbers of iterations of the three methods for the test problems. Taking account of the total number of iterations, Algorithm 1 outperformed the MCQN update with BFGS method on 11 problems (2,4,5,7,9,10,12,14,18,23,24). Additionally, Algorithm 1 outperformed the limited memeory BFGS method on 13 problems (1,2,3,7,9,12,15,16,18,19,20,21,23).…”

“…where F : R n → R n is a continuously differentiable mapping and the symmetry implies that the Jacobian F (x) satisfies F (x) = F (x) T . That symmetric nonlinear system has close relationships with many practical problems, such as the gradient mapping of unconstrained optimization problems, the Karush-Kuhn-Tuckrt (KKT) system of equality constrained optimization problem, the discretized two-point boundary value problem, and the saddle point problem (2) [1][2][3][4][5].…”

Scaled Three-Term Conjugate Gradient Methods for Solving Monotone Equations with Application

“…Additionally, whenever the Jacobian of F is symmetric, the system (1) is referred to as a symmetric system of nonlinear equations [5,6]. The symmetric system of nonlinear equations typically originates from the gradient mapping of unconstrained optimization problems and other relevant applications; more information on symmetric systems of equations can be found in [7,8], and the references therein. There are many methods or algorithms for solving the system of nonlinear equations in the literature.…”

A Modified Three-Term Conjugate Descent Derivative-Free Method for Constrained Nonlinear Monotone Equations and Signal Reconstruction Problems

“…However, they require computing and storing of the inverse Jacobian matrix at every iteration, which is not always easy to compute or not available due to singularity. As such, many researchers have suggested derivative-free approaches that do not require computation of the Jacobian matrix; among others are [16][17][18][19][20][21][22][23][24][25]. A modified three-term conjugate descent (CD) derivative-free method is of interest in this research.…”