A new filter algorithm for a system of nonlinear equations
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A new filter algorithm for a system of nonlinear equations Jueyu Wang1 · Chao Gu1 · Detong Zhu2 Received: 21 April 2020 / Revised: 12 July 2020 / Accepted: 10 August 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, we propose a new filter algorithm for the system of nonlinear equations. In previous work, the feasibility recovery phase is used to make up for the defect of the filter method, which requires a lot of calculation. To avoid it, the new algorithm is equipped with a new filter. At the same time, we discuss the division of the system of nonlinear equations so that the algorithm can achieve the best numerical results. The new algorithm switches to back tracking steps when a trial step produced by the trust region subproblem is unacceptable. Combining the new filter with both trust region and line search method, the convergence of the algorithm is obtained. The numerical results show the effectiveness of the new algorithm. Keywords Nonlinear equations · Trust region · Line search · Filter · Feasibility restoration phase · Convergence Mathematics Subject Classification 90C30 · 65K05
1 Introduction In this paper, we consider the system of nonlinear equations: ci (x) = 0, i = 1, 2, . . . , m, Rn
(1)
where x ∈ and ci (x) : → R for i = 1, 2, . . . , m. The problem (1) has many applications in engineering, such as nonlinear fitting, function approximating and parameter estimating. Many algorithms have been presented for solving the problem (1), for examples, Gauss–Newton methods (Bertsekas 1995; Levenberg 1944; Marquardt 1963; Nocedal and Wright 1999), Levenberg–Marquardt methods (Fan 2003; Nocedal and Wright 1999; Yamashita and Fukushima 2001; Zhang and Wang 2003), trust Rn
Communicated by Joerg Fliege.
B
Chao Gu [email protected]
1
School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai 201209, People’s Republic of China
2
Department of Mathematics, Shanghai Normal University, Shanghai 200234, People’s Republic of China 0123456789().: V,-vol
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region methods (Bahrami and Amini 2020; Conn et al. 2000; Wang 2015; Yuan 1998; Yuan and Sun 1997; Yuan et al. 2011), quasi-Newton method (Du and Gao 2009; Griewank 1986; Zhong and Deng 2002; Zhu 2005, 2006), inexact quasi-Newton method (Ariasa et al. 2020), derivative-free method (Begiato et al. 2020; Sharma and Arora 2016), Newton-SHSS method (Xie et al. 2019), filter methods (Gu 2011; Nie et al. 2008; Wang and Pu 2013). Nie et al. (2008) divides the set S = {1, 2, . . . , m} into S1 and S2 , where S2 denotes the complement S/S1 , and transformed (1) to a equivalent constrained optimization
min
ci2 (x)
(2a)
i∈S1
subject to
c j (x) = 0, j ∈ S2 .
(2b)
They define two merit functions ( i∈S1 ci2 (x) and C S2 (x)), and give a line search filter algorithm for the solution of (2). The numerical experiments show the effectiveness of the proposed algorithm. Subsequently, Gu (2011) studies a nonmonotone filter line se
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