A new black box method for monotone nonlinear equations

Abstract

This paper discusses a new derivative-free line search method for nonlinear monotone equations. It uses a derivative-free direction based on Dai–Liao method, along which an improved line search – called IDFLS – is tried. In contrast to the basic line search method – called BasicLS – by Solodov and Svaiter, IDFLS uses extrapolation steps to guarantee a decrease in the function norm. In fact, IDFLS never accept a point with the worst function norm, while BasicLS may accept such a point. The global convergence of our method is established under the assumption that the underlying mapping is monotone. The numerical results show that the new method is competitive in comparison with the state-of-the-art methods.

Keywords:

• Nonlinear monotone equations
• derivative-free optimization
• line search method
• global convergence

Acknowledgments

The first author was supported by the Petchra Pra Jom Klao Ph.D. Research Scholarship from King Mongkut's University of Technology Thonburi (Grant no. 16/2561). The second author acknowledges the financial support of the Doctoral Program Vienna Graduate School on Computational Optimization (VGSCO) funded by the Austrian Science Foundation under Project No W1260-N35. The authors also acknowledge the financial support provided by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT. Moreover, this project is funded by National Council of Thailand (NRCT) under Research Grants for Talented Mid-Career Researchers (Contract no. N41A640089).

Disclosure statement

No potential conflict of interest was reported by the authors.