A non-monotone pattern search approach for systems of nonlinear equations. Issue 1 (2nd January 2019)

• Record Type: Journal Article
• Authors: Amini, Keyvan; Kimiaei, Morteza; Khotanlou, Hassan
• Is Part Of: International journal of computer mathematics. Volume 96:Issue 1(2019)
• Page Start: 33
• Page End: 50
• Publication Date: 2019-01-02
• Subjects: Nonlinear equation -- pattern search -- coordinate search -- non-monotone technique -- theoretical convergence; 90C30 -- 93E24 -- 34A34; Computers -- Periodicals; Numerical analysis -- Periodicals; Automation -- Periodicals; 004.0151
• DOI: 10.1080/00207160.2017.1413552
• Languages: English
• ISSNs: 0020-7160

Abstract

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.