Globally convergent Jacobian-free nonlinear equation solvers based on non-monotone norm descent conditions and a modified line search technique

Abstract

A Jacobian-free nonlinear equation solver based on a search technique called a ‘spiral search’, which is a modification of the line search, is proposed in this paper. Under mild conditions, the solver is proved to be globally convergent. The method is then extended to an overdetermined system of nonlinear equations. Numerical results show that for some problems, the solver outperforms existing solvers based on the line search or the trust-region method.

Keywords:

• derivative-free methods
• modified line search
• methods of quasi-Newton type
• global convergence
• nonlinear equations

AMS Subject Classification :

• 90C30
• 90C53
• 90C56

Acknowledgements

The author would like to express his gratitude towards the anonymous referees for their careful reading and helpful and constructive comments.

Notes

This observation is based on the experience I and my colleague had while we were applying Li and Fukushima's algorithm to the waveform control problem of soft magnetic materials 16.

Note that is an artificial parameter that is introduced for the analysis of the algorithm and has nothing to do with the algorithm itself. Superficially, it may appear that a bad choice of causes a runaway of the backtracking, but in such cases, using a smaller resolves the difficulty.

The experiment of choosing the parameters of Table 1 and a parameter of NEWUOA were executed under the former environment, and all other experiments were executed under the latter.