scholarly journals An Improved Line Search Filter Method for the System of Nonlinear Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Zhong Jin ◽  
Yuqing Wang
Keyword(s):  
Global Convergence ◽  
Objective Function ◽  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Line Search ◽  
Equality Constraints ◽  
Filter Method ◽  
System Of Nonlinear Equations ◽  
Search Filter

An improved line search filter algorithm for the system of nonlinear equations is presented. We divide the equations into two groups, one contains the equations that are treated as equality constraints and the square of other equations is regarded as objective function. Two groups of equations are updated at every iteration in the works by Nie (2004, 2006, and 2006), by Nie et al. (2008), and by Gu (2011), while we just update them at the iterations when it is needed indeed. As a consequence, the scale of the calculation is decreased in a certain degree. Under some suitable conditions the global convergence can be induced. In the end, numerical experiments show that the method in this paper is effective.

scholarly journals A Nonmonotone Line Search Filter Algorithm for the System of Nonlinear Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Zhong Jin ◽  
Yuqing Wang
Keyword(s):  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Line Search ◽  
Filter Method ◽  
System Of Nonlinear Equations ◽  
Search Filter ◽  
Globally Convergent ◽  
Monotone Methods ◽  
Sufficient Reduction ◽  
New Iterative Method

We present a new iterative method based on the line search filter method with the nonmonotone strategy to solve the system of nonlinear equations. The equations are divided into two groups; some equations are treated as constraints and the others act as the objective function, and the two groups are just updated at the iterations where it is needed indeed. We employ the nonmonotone idea to the sufficient reduction conditions and filter technique which leads to a flexibility and acceptance behavior comparable to monotone methods. The new algorithm is shown to be globally convergent and numerical experiments demonstrate its effectiveness.

scholarly journals Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Huiping Cao
Keyword(s):  
Global Convergence ◽  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Large Scale ◽  
Line Search ◽  
Jacobian Matrix ◽  
Nonmonotone Line Search ◽  
Broyden’S Method ◽  
Large Scale Problems ◽  
Broyden's Method

Schubert’s method is an extension of Broyden’s method for solving sparse nonlinear equations, which can preserve the zero-nonzero structure defined by the sparse Jacobian matrix and can retain many good properties of Broyden’s method. In particular, Schubert’s method has been proved to be locally andq-superlinearly convergent. In this paper, we globalize Schubert’s method by using a nonmonotone line search. Under appropriate conditions, we show that the proposed algorithm converges globally and superlinearly. Some preliminary numerical experiments are presented, which demonstrate that our algorithm is effective for large-scale problems.

scholarly journals An Adaptive Nonmonotone Line Search Technique for Solving Systems of Nonlinear Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Masoud Hatamian ◽  
Mahmoud Paripour ◽  
Farajollah Mohammadi Yaghoobi ◽  
Nasrin Karamikabir
Keyword(s):  
Global Convergence ◽  
Nonlinear Equations ◽  
Line Search ◽  
Nonmonotone Line Search ◽  
Systems Of Nonlinear Equations ◽  
System Of Nonlinear Equations ◽  
Search Technique ◽  
Numerical Examples ◽  
Line Search Technique ◽  
Novel Algorithm

In this article, a new nonmonotone line search technique is proposed for solving a system of nonlinear equations. We attempt to answer this question how to control the degree of the nonmonotonicity of line search rules in order to reach a more efficient algorithm? Therefore, we present a novel algorithm that can avoid the increase of unsuccessful iterations. For this purpose, we show the robust behavior of the proposed algorithm by solving a few numerical examples. Under some suitable assumptions, the global convergence of our strategy is proved.

scholarly journals A line search filter approach for the system of nonlinear equations

2008 ◽  
Vol 55 (9) ◽  
pp. 2134-2141 ◽  
Author(s):  
Pu-yan Nie ◽  
Ming-yong Lai ◽  
Shu-jin Zhu ◽  
Pei-ai Zhang
Keyword(s):  
Nonlinear Equations ◽  
Line Search ◽  
System Of Nonlinear Equations ◽  
Search Filter ◽  
Filter Approach

A Non-Monotone Line Search Combination Technique for Unconstrained Optimization

2014 ◽  
Vol 8 (1) ◽  
pp. 218-221 ◽  
Author(s):  
Ping Hu ◽  
Zong-yao Wang
Keyword(s):  
Global Convergence ◽  
Unconstrained Optimization ◽  
Numerical Experiments ◽  
Line Search ◽  
Optimization Problems ◽  
Search Algorithm ◽  
New Combination ◽  
Combination Rule ◽  
Combination Technique ◽  
Unconstrained Optimization Problems

We propose a non-monotone line search combination rule for unconstrained optimization problems, the corresponding non-monotone search algorithm is established and its global convergence can be proved. Finally, we use some numerical experiments to illustrate the new combination of non-monotone search algorithm’s effectiveness.

scholarly journals A New Higher-Order Iterative Scheme for the Solutions of Nonlinear Systems

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 271 ◽  
Author(s):  
Ramandeep Behl ◽  
Ioannis K. Argyros
Keyword(s):  
Mathematical Modeling ◽  
Banach Space ◽  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Iterative Scheme ◽  
Real Life ◽  
System Of Nonlinear Equations ◽  
Life Problems ◽  
Lipschitz Constants ◽  
Better Than

Many real-life problems can be reduced to scalar and vectorial nonlinear equations by using mathematical modeling. In this paper, we introduce a new iterative family of the sixth-order for a system of nonlinear equations. In addition, we present analyses of their convergences, as well as the computable radii for the guaranteed convergence of them for Banach space valued operators and error bounds based on the Lipschitz constants. Moreover, we show the applicability of them to some real-life problems, such as kinematic syntheses, Bratu’s, Fisher’s, boundary value, and Hammerstein integral problems. We finally wind up on the ground of achieved numerical experiments, where they perform better than other competing schemes.

scholarly journals The “global” convergence of Broyden-like methods with suitable line search

1986 ◽  
Vol 28 (1) ◽  
pp. 75-92 ◽  
Author(s):  
Anderas Griewank
Keyword(s):  
Global Convergence ◽  
Iterative Methods ◽  
Level Set ◽  
Superlinear Convergence ◽  
Line Search ◽  
Divided Difference ◽  
Inverse Function ◽  
System Of Nonlinear Equations ◽  
Derivative Free ◽  
Quasi Newton

Iterative methods for solving a square system of nonlinear equations g(x) = 0 often require that the sum of squares residual γ (x) ≡ ½∥g(x)∥2 be reduced at each step. Since the gradient of γ depends on the Jacobian ∇g, this stabilization strategy is not easily implemented if only approximations Bk to ∇g are available. Therefore most quasi-Newton algorithms either include special updating steps or reset Bk to a divided difference estimate of ∇g whenever no satisfactory progress is made. Here the need for such back-up devices is avoided by a derivative-free line search in the range of g. Assuming that the Bk are generated from an rbitrary B0 by fixed scale updates, we establish superlinear convergence from within any compact level set of γ on which g has a differentiable inverse function g−1.

scholarly journals Finding the Roots of System of Nonlinear Equations by a Novel Filled Function Method

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Wei-Xiang Wang ◽  
You-Lin Shang ◽  
Wei-Gang Sun ◽  
Ying Zhang
Keyword(s):  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Function Method ◽  
Optimization Problem ◽  
Test Problems ◽  
System Of Nonlinear Equations ◽  
Global Optimization Problem ◽  
Filled Function ◽  
Filled Function Method ◽  
Constrained System

We present a novel filled function approach to solve box-constrained system of nonlinear equations. The system is first transformed into an equivalent nonsmooth global minimization problem, and then a new filled function method is proposed to solve this global optimization problem. Numerical experiments on several test problems are conducted and the computational results are also reported.

scholarly journals A Modified Hestenes-Stiefel-Type Derivative-Free Method for Large-Scale Nonlinear Monotone Equations

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 168 ◽  
Author(s):  
Zhifeng Dai ◽  
Huan Zhu
Keyword(s):  
Global Convergence ◽  
Large Scale ◽  
Line Search ◽  
System Of Nonlinear Equations ◽  
Inexact Newton Method ◽  
Monotone Equations ◽  
Smoothing Methods ◽  
Derivative Free ◽  
Derivative Free Method ◽  
Nonlinear Monotone Equations

The goal of this paper is to extend the modified Hestenes-Stiefel method to solve large-scale nonlinear monotone equations. The method is presented by combining the hyperplane projection method (Solodov, M.V.; Svaiter, B.F. A globally convergent inexact Newton method for systems of monotone equations, in: M. Fukushima, L. Qi (Eds.)Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Kluwer Academic Publishers. 1998, 355-369) and the modified Hestenes-Stiefel method in Dai and Wen (Dai, Z.; Wen, F. Global convergence of a modified Hestenes-Stiefel nonlinear conjugate gradient method with Armijo line search. Numer Algor. 2012, 59, 79-93). In addition, we propose a new line search for the derivative-free method. Global convergence of the proposed method is established if the system of nonlinear equations are Lipschitz continuous and monotone. Preliminary numerical results are given to test the effectiveness of the proposed method.

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