• Previous Article
    An outcome space algorithm for minimizing the product of two convex functions over a convex set
  • JIMO Home
  • This Issue
  • Next Article
    Applications of a nonlinear optimization solver and two-stage comprehensive Denoising techniques for optimum underwater wideband sonar echolocation system
January  2013, 9(1): 227-241. doi: 10.3934/jimo.2013.9.227

On the Levenberg-Marquardt methods for convex constrained nonlinear equations

1. 

Department of Mathematics, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China

Received  January 2012 Revised  July 2012 Published  December 2012

In this paper, both the constrained Levenberg-Marquardt method and the projected Levenberg-Marquardt method are presented for nonlinear equations $F(x)=0$ subject to $x\in X$, where $X$ is a nonempty closed convex set. The Levenberg-Marquardt parameter is taken as $\| F(x_k) \|_2^\delta$ with $\delta\in (0, 2]$. Under the local error bound condition which is weaker than nonsingularity, the methods are shown to have the same convergence rate, which includes not only the convergence results obtained in [12] for $\delta=2$ but also the results given in [7] for unconstrained nonlinear equations.
Citation: Jinyan Fan. On the Levenberg-Marquardt methods for convex constrained nonlinear equations. Journal of Industrial & Management Optimization, 2013, 9 (1) : 227-241. doi: 10.3934/jimo.2013.9.227
References:
[1]

Appl. Numer. Math., 44 (2003), 257-280. doi: 10.1016/S0168-9274(02)00170-8.  Google Scholar

[2]

Optim. Methods Software, 20 (2005), 1-22.  Google Scholar

[3]

Optim. Meth. Software, 17 (2002), 605-626.  Google Scholar

[4]

Prentice-Hall, Englewood Cliffs, NJ, 1983.  Google Scholar

[5]

Optim. Meth. Software, 5 (1995), 319-345. doi: 10.1080/10556789508805619.  Google Scholar

[6]

IEEE Service Center, Piscataway, New Jersey, 1996. Google Scholar

[7]

Computational Optimization and Applications, 34 (2006), 47-62.  Google Scholar

[8]

Computing, 74 (2005), 23-39. doi: 10.1007/s00607-004-0083-1.  Google Scholar

[9]

Vol. 33, Kluwer Academic Publishers, The Netherlands, 1999.  Google Scholar

[10]

Lecture Notes in Economics and Mathematical Systems, Vol. 187, Springer, New York, 1981.  Google Scholar

[11]

in "Complementarity: Applications, Algorithms and Extensions"(M. C. Ferris, O. L. Mangasarian and J. S. Pang eds.), Kluwer Academic, Dordrecht, 2001, pp 179-200.  Google Scholar

[12]

Journal of Computational and Applied Mathematics, 173 (2005), 321-343.  Google Scholar

[13]

SIAM, Philadelphia, 1995.  Google Scholar

[14]

Comput. Appl. Math., 16 (1997), 215-235.  Google Scholar

[15]

Quart. Appl. Math., 2 (1944), 164-166.  Google Scholar

[16]

SIAM J. Appl. Math., 11 (1963), 431-441.  Google Scholar

[17]

Appl. Math. Comput., 22 (1987), 333-361.  Google Scholar

[18]

ACM Trans. Math. Software, 16 (1990), 143-151. Google Scholar

[19]

SIAM J. Optim., 9 (1999), 729-754. doi: 10.1137/S1052623497318980.  Google Scholar

[20]

in "Lecture Notes in Mathematics 630: Numerical Analysis"(G. A. Watson ed.), Springer-Verlag, Berlin, 1978, pp. 105-116.  Google Scholar

[21]

Academic Press, New York, 1970.  Google Scholar

[22]

Journal of Optimization Theories and Applications, 120 (2004), 601-649.  Google Scholar

[23]

SIAM J. Optim., 11 (2001), 889-917.  Google Scholar

[24]

Math. Programming, 74 (1996), 159-195.  Google Scholar

[25]

John Wiley and Sons, New York, NY, 1996. Google Scholar

[26]

Computing (Supplement 15), (2001), 237-249. Google Scholar

[27]

Numerical Algebra, Control and Optimization, 1 (2011), 15-34.  Google Scholar

show all references

References:
[1]

Appl. Numer. Math., 44 (2003), 257-280. doi: 10.1016/S0168-9274(02)00170-8.  Google Scholar

[2]

Optim. Methods Software, 20 (2005), 1-22.  Google Scholar

[3]

Optim. Meth. Software, 17 (2002), 605-626.  Google Scholar

[4]

Prentice-Hall, Englewood Cliffs, NJ, 1983.  Google Scholar

[5]

Optim. Meth. Software, 5 (1995), 319-345. doi: 10.1080/10556789508805619.  Google Scholar

[6]

IEEE Service Center, Piscataway, New Jersey, 1996. Google Scholar

[7]

Computational Optimization and Applications, 34 (2006), 47-62.  Google Scholar

[8]

Computing, 74 (2005), 23-39. doi: 10.1007/s00607-004-0083-1.  Google Scholar

[9]

Vol. 33, Kluwer Academic Publishers, The Netherlands, 1999.  Google Scholar

[10]

Lecture Notes in Economics and Mathematical Systems, Vol. 187, Springer, New York, 1981.  Google Scholar

[11]

in "Complementarity: Applications, Algorithms and Extensions"(M. C. Ferris, O. L. Mangasarian and J. S. Pang eds.), Kluwer Academic, Dordrecht, 2001, pp 179-200.  Google Scholar

[12]

Journal of Computational and Applied Mathematics, 173 (2005), 321-343.  Google Scholar

[13]

SIAM, Philadelphia, 1995.  Google Scholar

[14]

Comput. Appl. Math., 16 (1997), 215-235.  Google Scholar

[15]

Quart. Appl. Math., 2 (1944), 164-166.  Google Scholar

[16]

SIAM J. Appl. Math., 11 (1963), 431-441.  Google Scholar

[17]

Appl. Math. Comput., 22 (1987), 333-361.  Google Scholar

[18]

ACM Trans. Math. Software, 16 (1990), 143-151. Google Scholar

[19]

SIAM J. Optim., 9 (1999), 729-754. doi: 10.1137/S1052623497318980.  Google Scholar

[20]

in "Lecture Notes in Mathematics 630: Numerical Analysis"(G. A. Watson ed.), Springer-Verlag, Berlin, 1978, pp. 105-116.  Google Scholar

[21]

Academic Press, New York, 1970.  Google Scholar

[22]

Journal of Optimization Theories and Applications, 120 (2004), 601-649.  Google Scholar

[23]

SIAM J. Optim., 11 (2001), 889-917.  Google Scholar

[24]

Math. Programming, 74 (1996), 159-195.  Google Scholar

[25]

John Wiley and Sons, New York, NY, 1996. Google Scholar

[26]

Computing (Supplement 15), (2001), 237-249. Google Scholar

[27]

Numerical Algebra, Control and Optimization, 1 (2011), 15-34.  Google Scholar

[1]

Haiyan Wang, Jinyan Fan. Convergence properties of inexact Levenberg-Marquardt method under Hölderian local error bound. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2265-2275. doi: 10.3934/jimo.2020068

[2]

Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185

[3]

Mohsen Abdolhosseinzadeh, Mir Mohammad Alipour. Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 321-332. doi: 10.3934/naco.2020028

[4]

Tobias Breiten, Sergey Dolgov, Martin Stoll. Solving differential Riccati equations: A nonlinear space-time method using tensor trains. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 407-429. doi: 10.3934/naco.2020034

[5]

Xiaofei Liu, Yong Wang. Weakening convergence conditions of a potential reduction method for tensor complementarity problems. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021080

[6]

Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023

[7]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327

[8]

Yaonan Ma, Li-Zhi Liao. The Glowinski–Le Tallec splitting method revisited: A general convergence and convergence rate analysis. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1681-1711. doi: 10.3934/jimo.2020040

[9]

Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017

[10]

Woocheol Choi, Youngwoo Koh. On the splitting method for the nonlinear Schrödinger equation with initial data in $ H^1 $. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3837-3867. doi: 10.3934/dcds.2021019

[11]

Jie-Wen He, Chi-Chon Lei, Chen-Yang Shi, Seak-Weng Vong. An inexact alternating direction method of multipliers for a kind of nonlinear complementarity problems. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 353-362. doi: 10.3934/naco.2020030

[12]

Li Chu, Bo Wang, Jie Zhang, Hong-Wei Zhang. Convergence analysis of a smoothing SAA method for a stochastic mathematical program with second-order cone complementarity constraints. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1863-1886. doi: 10.3934/jimo.2020050

[13]

Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018

[14]

Kai Kang, Taotao Lu, Jing Zhang. Financing strategy selection and coordination considering risk aversion in a capital-constrained supply chain. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021042

[15]

Maolin Cheng, Yun Liu, Jianuo Li, Bin Liu. Nonlinear Grey Bernoulli model NGBM (1, 1)'s parameter optimisation method and model application. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021054

[16]

Abderrazak Chrifi, Mostafa Abounouh, Hassan Al Moatassime. Galerkin method of weakly damped cubic nonlinear Schrödinger with Dirac impurity, and artificial boundary condition in a half-line. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021030

[17]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[18]

Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3119-3142. doi: 10.3934/dcdsb.2020222

[19]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[20]

Antonio De Rosa, Domenico Angelo La Manna. A non local approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021059

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (122)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]