Abstract
In this paper, we propose a nonmonotone adap-tive trust-region method for solving symmetric nonlinear equations problems. The convergent result of the presented method will be estab-lished under favorable conditions. Numerical results are reported.
Highlights
INTRODUCTIONWhere g : Rn Rn is continuously differentiable, the Jacobian g(x) of g is symmetric for all x Rn
Consider the following system of nonlinear equations: g(x) 0, x Rn (1)where g : Rn Rn is continuously differentiable, the Jacobian g(x) of g is symmetric for all x Rn .Define a norm function by (x) || g(x) ||2It is not difficult to see that the nonlinear equations problemEq.1 is equivalent to the following global optimization problem min (x), x Rn (2)Here and throughout this paper, we use the following notations.Bk is a symmetric matrix which is an approxima-tion of g(x)T g(x) .It is well known that there are many methods for the unconstrained optimization problem minx Rn f (x), where the trust-region methods are very successful, e.g., Moré and Sorensen [8]
Gauss-Newton-based BFGS method is proposed by Li and Fukushima [31] for solving symmetric nonlinear equations
Summary
Where g : Rn Rn is continuously differentiable, the Jacobian g(x) of g is symmetric for all x Rn. A. Gauss-Newton-based BFGS method is proposed by Li and Fukushima [31] for solving symmetric nonlinear equations. Gauss-Newton-based BFGS method is proposed by Li and Fukushima [31] for solving symmetric nonlinear equations Inspired by their ideas, Wei [32] and Yuan [33,34,35,36,37] made a further study. Presented a new backtracking inexact BFGS method for symmetric nonlinear equations. Inspired by the technique of Zhang [23], we propose a new nonmotone adaptive trust region method for solving. Where (xk ) g(xk )g(xk ) , k cp || (xk ) || Mk , 0 c 1, M k || Bk 1 ||, p is a nonnegative integer, and matrix Bk is an approximation of g(xk )T g(xk ) which is generated by the following BFGS formula [31]: Bk 1.
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