Three derivative-free projection methods for nonlinear equations with convex constraints

2014 ◽  
Vol 47 (1-2) ◽  
pp. 265-276 ◽  
Author(s):  
Min Sun ◽  
Jing Liu
Keyword(s):  
Nonlinear Equations ◽  
Projection Methods ◽  
Convex Constraints ◽  
Derivative Free

scholarly journals A unified derivative-free projection method model for large-scale nonlinear equations with convex constraints

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yigui Ou ◽  
Wenjie Xu
Keyword(s):  
Nonlinear Equations ◽  
Projection Method ◽  
Large Scale ◽  
Practical Method ◽  
Projection Methods ◽  
Convex Constraints ◽  
Derivative Free ◽  
Constrained Equations ◽  
Practical Algorithm ◽  
Method Model

<p style='text-indent:20px;'>Motivated by recent derivative-free projection methods proposed in the literature for solving nonlinear constrained equations, in this paper we propose a unified derivative-free projection method model for large-scale nonlinear equations with convex constraints. Under mild conditions, the global convergence and convergence rate of the proposed method are established. In order to verify the feasibility and effectiveness of the model, a practical algorithm is devised and the corresponding numerical experiments are reported, which show that the proposed practical method is efficient and can be applied to solve large-scale nonsmooth equations. Moreover, the proposed practical algorithm is also extended to solve the obstacle problem.</p>

scholarly journals AN EFFICIENT HYBRID DERIVATIVE-FREE PROJECTION ALGORITHM FOR CONSTRAINT NONLINEAR EQUATIONS

2021 ◽  
Vol 40 (3) ◽  
pp. 64-75
Author(s):  
Kanikar Muangchoo
Keyword(s):  
Nonlinear Equations ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Projection Algorithm ◽  
Monotonicity Condition ◽  
Convex Constraints ◽  
Sufficient Descent Condition ◽  
Lipschitz Continuous ◽  
Derivative Free

In this paper, by combining the Solodov and Svaiter projection technique with the conjugate gradient method for unconstrained optimization proposed by Mohamed et al. (2020), we develop a derivative-free conjugate gradient method to solve nonlinear equations with convex constraints. The proposed method involves a spectral parameter which satisfies the sufficient descent condition. The global convergence is proved under the assumption that the underlying mapping is Lipschitz continuous and satisfies a weaker monotonicity condition. Numerical experiment shows that the proposed method is efficient.

scholarly journals Projection method with inertial step for nonlinear equations: Application to signal recovery

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Abdulkarim Hassan Ibrahim ◽  
Poom Kumam ◽  
Min Sun ◽  
Parin Chaipunya ◽  
Auwal Bala Abubakar
Keyword(s):  
Nonlinear Equations ◽  
Sparse Signal ◽  
Signal Recovery ◽  
Convex Constraints ◽  
Lipschitz Continuous ◽  
Convergence Results ◽  
Derivative Free ◽  
Derivative Free Method ◽  
Inertial Extrapolation ◽  
Sensing Performance

<p style='text-indent:20px;'>In this paper, using the concept of inertial extrapolation, we introduce a globally convergent inertial extrapolation method for solving nonlinear equations with convex constraints for which the underlying mapping is monotone and Lipschitz continuous. The method can be viewed as a combination of the efficient three-term derivative-free method of Gao and He [Calcolo. 55(4), 1-17, 2018] with the inertial extrapolation step. Moreover, the algorithm is designed such that at every iteration, the method is free from derivative evaluations. Under standard assumptions, we establish the global convergence results for the proposed method. Numerical implementations illustrate the performance and advantage of this new method. Moreover, we also extend this method to solve the LASSO problems to decode a sparse signal in compressive sensing. Performance comparisons illustrate the effectiveness and competitiveness of our algorithm.</p>

scholarly journals An Efficient Conjugate Gradient Method for Convex Constrained Monotone Nonlinear Equations with Applications

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 767 ◽  
Author(s):  
Abubakar ◽  
Kumam ◽  
Mohammad ◽  
Awwal
Keyword(s):  
Nonlinear Equations ◽  
Line Search ◽  
Search Strategy ◽  
Convex Constraints ◽  
Specific Direction ◽  
Derivative Free ◽  
Constraint Set ◽  
Derivative Free Method ◽  
Initial Iterate ◽  
Closed Convex Set

This research paper proposes a derivative-free method for solving systems of nonlinearequations with closed and convex constraints, where the functions under consideration are continuousand monotone. Given an initial iterate, the process first generates a specific direction and then employsa line search strategy along the direction to calculate a new iterate. If the new iterate solves theproblem, the process will stop. Otherwise, the projection of the new iterate onto the closed convex set(constraint set) determines the next iterate. In addition, the direction satisfies the sufficient descentcondition and the global convergence of the method is established under suitable assumptions.Finally, some numerical experiments were presented to show the performance of the proposedmethod in solving nonlinear equations and its application in image recovery problems.

scholarly journals On a Derivative-Free Variant of King’s Family with Memory

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
M. Sharifi ◽  
S. Karimi Vanani ◽  
F. Khaksar Haghani ◽  
M. Arab ◽  
S. Shateyi
Keyword(s):  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Order Of Convergence ◽  
Derivative Free ◽  
With Memory

The aim of this paper is to construct a method with memory according to King’s family of methods without memory for nonlinear equations. It is proved that the proposed method possesses higherR-order of convergence using the same number of functional evaluations as King’s family. Numerical experiments are given to illustrate the performance of the constructed scheme.

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