A modified nonmonotone BFGS algorithm for solving smooth nonlinear equations

Optimization Letters

Volumn 8, Issue 6, 2014, Pages 1845-1860

  Wan, Zhong ^a   Chen, Yu ^a   Huang, Shuai ^a   Feng, Dongdong ^a  

^a CENTRAL SOUTH UNIVERSITY   (China)

Author keywords

Global convergence; Modified BFGS algorithm; Nonmonotone technique; Smooth nonlinear equations

Indexed keywords

BENCHMARKING; ITERATIVE METHODS; NONLINEAR EQUATIONS; SEARCH ENGINES; TWO PHASE FLOW;

ALGORITHMIC PARAMETERS; BENCHMARK TEST PROBLEM; GLOBAL CONVER-GENCE; MODIFIED BFGS ALGORITHM; NONLINEAR SYSTEM OF EQUATIONS; NONMONOTONE LINE SEARCH; NONMONOTONE TECHNIQUE; SYSTEM OF NONLINEAR EQUATIONS;

ALGORITHMS;

EID: 84905106012     PISSN: 18624472     EISSN: 18624480     Source Type: Journal    
DOI: 10.1007/s11590-013-0678-6     Document Type: Article

References (20)

1. Gu, G. Z., Li, D. H., Qi, L. Q., Zhou, S. Z.: Descent directions of quasi-Newton methods for symmetric nonlinear equations. SIAM J. Numer. Anal. 40, 1763-1774 (2003).
2. Yuan, G. L., Liu, X. W.: A new backtracking inexact BFGS method for symmetric nonlinear equations. Comput. Math. Appl. 55, 116-129 (2009).
3. Guo, Q., Liu, J. G., Wang, D. H.: A modified BFGS method and its superlinear convergence in nonconvex minimization with general line search rule. J. Appl. Math. Comput. 28, 435-446 (2008).
4. Guo, Q., Liu, J. G.: Global convergence of a modified BFGS-type method for unconstrained nonconvex minimization. J. Appl. Math. Comput. 24(1-2), 325-331 (2007).
5. Li, D. H., Fukushima, M.: On the global convergence of the BFGS method for nonconvex unconstrained optimization problems. SIAM J. Optim. 11(4), 1054-1064 (2001).
6. Li, D. H., Fukushima, M.: A globally and superlinearly convergent Gauss-Newton-based BFGS methods for symmetric nonlinear equations. SIAM J. Numer. Anal. 37, 152-172 (1999).
7. Li, D. H., Cheng, W. Y.: Recent progress in global convergence of quasi-Newton methods for nonlinear equations. Hokkaido Math. J. 36, 729-743 (2007).
8. Liu, J. G., Guo, Q.: Global convergence properties of the modified BFGS method associating with general line search model. J. Appl. Math. Comput. 18(1-2), 195-205 (2004).
9. Zhou, W. J., Zhang, L.: Global convergence of the nonmonotone MBFGS method for nonconvex unconstrained minimization. J. Comput. Appl. Math. 223, 40-47 (2009).
10. Birgin, E. G., Krejic, N., Martínez, J. M.: Globally convergent inexact quasi-Newton methods for solving nonlinear systems. Numer. Algorithms 32, 249-260 (2003).
11. Kelley, C. T.: Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia (1995).
12. Martínez, J. M.: Practical quasi-Newton methods for solving nonlinear systems. J. Comput. Appl. Math. 124, 97-122 (2000).
13. Grippo, L., Lampariello, F., Lucidi, S.: Newton-type algorithms with nonmonotone line search for large-scale unconstrained optimization. Syst. Model. Optim. 113, 187-196 (1988).
14. Grippo, L., Lampariello, F., Lucidi, S.: A class of nonmonotone stabilization methods in unconstrained optimization. Numerische Mathematik, vol 59, pp. 779-805 (1991).
15. Xiao, Y. H., Sun, H. J., Wang, Z. G.: A globally convergent BFGS method with nonmonotone line search for non-convex minimization. J. Comput. Appl. Math. 230, 95-106 (2009).
16. Yuan, G. L., Wei, Z. X.: The superlinear convergence analysis of a nonmonotone BFGS algorithm on convex objective functions. Acta Mathematica Sinica (English Series) 24(1), 35-42 (2008).
17. Zhu, D. T.: Nonmonotone backtracking inexact quasi-Newton algorithms for solving smooth nonlinear equations. Appl. Math. Comput. 161, 875-895 (2005).
18. Zhang, H. C., Hager, W. W.: A nonmonotone line search technique and its application to unconstrained optimization. SIAM J. Optim. 14(4), 1043-1056 (2004).
19. Shi, Z. J., Shen, J.: Convergence of nonmonotone line search method. J. Comput. Appl. Math. 193, 397-412 (2006).
20. Luksan, L.: Inexact trust region method for large sparse systems of nonlinear equations. Comput. Optim. Appl. 81(3), 569-590 (1994).