Multivariate spectral DY-type projection method for convex constrained nonlinear monotone equations.

*(English)*Zbl 1368.49038Summary: In this paper, we consider a multivariate spectral DY-type projection method for solving nonlinear monotone equations with convex constraints. The search direction of the proposed method combines those of the multivariate spectral gradient method and DY conjugate gradient method. With no need for the derivative information, the proposed method is very suitable to solve large-scale nonsmooth monotone equations. Under appropriate conditions, we prove the global convergence and R-linear convergence rate of the proposed method. The preliminary numerical results also indicate that the proposed method is robust and effective.

##### MSC:

49M37 | Numerical methods based on nonlinear programming |

65H10 | Numerical computation of solutions to systems of equations |

65K05 | Numerical mathematical programming methods |

##### Keywords:

nonlinear monotone equations; multivariate spectral gradient method; DY conjugate gradient method; global convergence##### Software:

MCPLIB
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\textit{J. Liu} and \textit{S. Li}, J. Ind. Manag. Optim. 13, No. 1, 283--295 (2017; Zbl 1368.49038)

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##### References:

[1] | J. M. Barizilai, Two point step size gradient methods,, IMA Journal on Numerical Analysis, 8, 141, (1988) · Zbl 0638.65055 |

[2] | H. H. Bauschke, A weak-to-strong convergence principle for Fejèer-monotone methods in Hilbert spaces,, Mathematical Methods and Operations Research, 26, 248, (2001) · Zbl 1082.65058 |

[3] | S. Bellavia, A globally convergent Newton-GMRES subspace method for systems of nonlinear equations,, SIAM Journal on Scientific Computing, 23, 940, (2001) · Zbl 0998.65053 |

[4] | Y. H. Dai, A nonlinear conjugate gradient with a strong global convergence property,, SIAM Journal on Optimization, 10, 177, (1999) · Zbl 0957.65061 |

[5] | J. E. Dennis, A characterization of superlinear convergence and its application to quasi-Newton methods,, Mathematics of Computation, 28, 549, (1974) · Zbl 0282.65042 |

[6] | J. E. Dennis, Quasi-Newton method, motivation and theory,, SIAM Review, 19, 46, (1997) · Zbl 0356.65041 |

[7] | S. P. Dirkse, MCPLIB: A collection of nonlinear mixed complementarity problems,, Optimization Methods and Software, 5, 319, (1995) |

[8] | L. Han, Multivariate spectral gradient method for unconstrained optimization,, Applied Mathematics and Computation, 201, 621, (2008) · Zbl 1155.65046 |

[9] | A. N. Iusem, Newton-type methods with generalized distances for constrained optmization,, Optimization, 41, 257, (1997) · Zbl 0905.49015 |

[10] | W. La Cruz, Nonmonotone spectral methods for large-scale nonlinear systems,, Optimization Methods and Software, 18, 583, (2003) · Zbl 1069.65056 |

[11] | W. La Cruz, Spectral residual method without gradient minformation for solving large-scale nonlinear systems of equations,, Mathematics of Computation, 75, 1429, (2006) · Zbl 1122.65049 |

[12] | Q. N. Li, A class of derivative-free methods for large-scale nonlinear monotone equations,, IMA Journal on Numerical Analysis, 31, 1625, (2011) · Zbl 1241.65047 |

[13] | K. Meintjes, A methodology for solving chemical equilibrium systems,, Applied Mathematics and Computation, 22, 333, (1987) · Zbl 0616.65057 |

[14] | K. Meintjes, Chemical equilibrium systems as numerical test problems,, ACM Transactions on Mathematical Software, 16, 143, (1990) · Zbl 0900.65153 |

[15] | L. Qi, A nonsmooth version of Newton’s method,, Mathematical Programming, 58, 353, (1999) · Zbl 0780.90090 |

[16] | M. V. Solodov, A globally convergent inexact Newton method for systems of monotone equations,, in Reformulation: Nonsmooth, 355, (1999) · Zbl 0928.65059 |

[17] | C. W. Wang, A projection method for a system of nonlinear monotone equations with convex constraints,, Mathematical Methods and Operations Research, 66, 33, (2007) · Zbl 1126.90067 |

[18] | A. J. Wood, <em>Power Generations Operations and Control</em>,, Wiley, (1996) |

[19] | N. Yamashita, On the rate of convergence of the Levenberg-Marquardt method,, Computing, 15, 239, (2001) · Zbl 1001.65047 |

[20] | N. Yamashita, Modified Newton methods for sovling a semismooth reformulation of monotone complementary problems,, Mathematical Programming, 76, 469, (1997) · Zbl 0872.90102 |

[21] | Z. S. Yu, A multivariate spectral projected gradient method for bound constrained optimization,, Journal of Computational and Applied Mathematics, 235, 2263, (2011) · Zbl 1209.65065 |

[22] | G. H. Yu, Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints,, Journal of Industrial and Management Optimization, 9, 117, (2013) · Zbl 1264.49037 |

[23] | L. Zhang, Spectral gradient projection method for solving nonlinear monotone equations,, Journal of Computational and Applied Mathematics, 196, 478, (2006) · Zbl 1128.65034 |

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