A Perry-type derivative-free algorithm for solving nonlinear system of equations and minimizing ℓ₁ regularized problem

Abstract

In this paper, we propose a Perry-type derivative-free algorithm for solving systems of nonlinear equations. The algorithm is based on the well-known BFGS quasi-Newton method with a modified Perry's parameter. The global convergence of the algorithm is established without assumption on the regularity or boundedness of the solution set. Meanwhile, the sequence of iterates generated by the algorithm converges globally to the solution of the problem provided that the function is Lipschitz continuous and monotone. Preliminary numerical experiments on some collection of general nonlinear equations and convex constrained nonlinear monotone equations demonstrate the efficiency of the algorithm. Moreover, we successfully apply the proposed algorithm to solve signal recovery problem.

Keywords:

• Derivative-free method
• nonlinear equations
• projection method
• signal recovery

2010 Mathematics Subject Classifications:

• 90C30
• 90C06
• 90C56

Acknowledgments

The authors thank the two anonymous reviewers for their useful comments and suggestions which greatly improved the manuscript. The authors acknowledge the financial support provided by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT. The first author was supported by the ‘Petchra Pra Jom Klao Ph.D. Research Scholarship from King Mongkut's University of Technology Thonburi’ (Contract No. 42/2560).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors acknowledge the financial support provided by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT. The first author was supported by the ‘Petchra Pra Jom Klao Ph.D. Research Scholarship from King Mongkut's University of Technology Thonburi’ (Contract No. 42/2560).