A non-monotone pattern search approach for systems of nonlinear equations

ABSTRACT

In this paper, a new pattern search is proposed to solve the systems of nonlinear equations. We introduce a new non-monotone strategy which includes a convex combination of the maximum function of some preceding successful iterates and the current function. First, we produce a stronger non-monotone strategy in relation to the generated strategy by Gasparo et al. [Nonmonotone algorithms for pattern search methods, Numer. Algorithms 28 (2001), pp. 171–186] whenever iterates are far away from the optimizer. Second, when iterates are near the optimizer, we produce a weaker non-monotone strategy with respect to the generated strategy by Ahookhosh and Amini [An efficient nonmonotone trust-region method for unconstrained optimization, Numer. Algorithms 59 (2012), pp. 523–540]. Third, whenever iterates are neither near the optimizer nor far away from it, we produce a medium non-monotone strategy which will be laid between the generated strategy by Gasparo et al. [Nonmonotone algorithms for pattern search methods, Numer. Algorithms 28 (2001), pp. 171–186] and Ahookhosh and Amini [An efficient nonmonotone trust-region method for unconstrained optimization, Numer. Algorithms 59 (2012), pp. 523–540]. Reported are numerical results of the proposed algorithm for which the global convergence is established.

Keywords:

• Nonlinear equation
• pattern search
• coordinate search
• non-monotone technique
• theoretical convergence

2010 AMS Subject Classifications:

• 90C30
• 93E24
• 34A34

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The second author acknowledges the financial support of the Doctoral Program ‘Vienna Graduate School on Computational Optimization’ funded by Austrian Science Foundation under Project No W1260-N35.