A nonmonotone smoothing Newton algorithm for solving general box constrained variational inequalities.

*(English)*Zbl 1402.90193Summary: In this paper, based on a new smoothing function, the general box constrained variational inequalities are solved by a smoothing Newton algorithm with a nonmonotone line search. The proposed algorithm is proved to be globally and locally superlinearly convergent under suitable assumptions. The preliminary numerical results are reported.

##### MSC:

90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |

90C53 | Methods of quasi-Newton type |

##### Keywords:

general box constrained variational inequalities; smoothing Newton method; smoothing function; convergence; nonmonotone line search##### Software:

MCPLIB
PDF
BibTeX
Cite

\textit{N. Zhao} and \textit{T. Ni}, Optimization 67, No. 8, 1231--1245 (2018; Zbl 1402.90193)

Full Text:
DOI

##### References:

[1] | Chan, D; Pang, JS, The generalized quasi-variational inequality problem, Math Oper Res, 7, 211-222, (1982) · Zbl 0502.90080 |

[2] | Chen, B; Chen, X, A global linear and local quadratic continuation smoothing method for variational inequalities with box constraints, Comput Optim Appl, 13, 131-158, (2000) · Zbl 0987.90079 |

[3] | Chen, X; Qi, L; Sun, D, Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities, Math Comput, 67, 519-540, (1998) · Zbl 0894.90143 |

[4] | Chen, X; Ye, Y, On homotopy-smoothing methods for box-constrained variational inequalities, SIAM J Control Optim, 37, 589-616, (1999) · Zbl 0973.65051 |

[5] | Facchinei, F; Pang, JS, Finite-dimensional variational inequalities and complementarity problems, (2003), Springer Science & Business Media, New York |

[6] | Facchinei, F; Kanzow, C; Sagratella, S, Solving quasi-variational inequalities via their KKT-conditions, Math Program, 144, 369-412, (2014) · Zbl 1293.65100 |

[7] | He, BS, Inexact implicit methods for monotone general variational inequalities, Math Program, 86, 199-217, (1999) · Zbl 0979.49006 |

[8] | Kanzow, C; Fukushima, M, Theoretical and numerical investigation of the D-gap function for box constrained variational inequalities, Math Program, 83, 55-87, (1998) · Zbl 0920.90134 |

[9] | Li, M; Liao, LZ; Yuan, XM, Proximal point algorithms for general variational inequalities, J Optim Theory Appl, 142, 125-145, (2009) · Zbl 1198.90374 |

[10] | Noor, MA, General variational inequalities, Appl Math Lett, 1, 119-122, (1988) · Zbl 0655.49005 |

[11] | Ou, YG; Lin, HC, A continuous method model for solving general variational inequality, Int J Comput Math, 93, 1899-1920, (2015) · Zbl 1355.65084 |

[12] | Qi, L; Sun, D; Zhou, G, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequality problems, Math Program, 87, 1-35, (2000) · Zbl 0989.90124 |

[13] | Wang, XB; Ma, CF; Li, MY, A globally and superlinearly convergent quasi-Newton method for general box constrained variational inequalities without smoothing approximation, J Global Optim, 50, 675-694, (2011) · Zbl 1254.90261 |

[14] | Ferris, M; Kanzow, C, Feasible descent algorithms for mixed complentarity problems, Math Program, 86, 475-497, (1999) · Zbl 0946.90094 |

[15] | Li, D; Fukushima, M, Smoothing Newton and quasi-Newton methods for mixed complementarity problems, Comput Optim Appl, 17, 203-230, (2000) · Zbl 1168.90623 |

[16] | Zhao, N; Huang, ZH, A nonmonotone smoothing Newton algorithm for solving box constrained variational inequalities with a P_{0} function, J Indus Manage Optim, 7, 467-482, (2011) · Zbl 1225.90140 |

[17] | Tawhid, MA, An application of H-differentiability to nonnegative and unrestricted generalized complementarity, Comput Optim Appl, 39, 51-74, (2008) · Zbl 1147.90405 |

[18] | Huang, ZH; Han, JY; Xu, DC; Zhang, LP, The non-interior continuation methods for solving the P\textsubscript{0} function non-linear complementarity problem, Sci China, 44, 1107-1114, (2001) · Zbl 1002.90072 |

[19] | Zhang, HC; Hager, WW, A nonmonotone line search technique and its application to unconstrained optimization, SIAM J Optim, 14, 1043-1056, (2004) · Zbl 1073.90024 |

[20] | Hu, SL; Huang, ZH; Lu, N, A non-monotone line search algorithm for unconstrained optimization, J Sci Comput, 42, 38-53, (2010) · Zbl 1203.90146 |

[21] | Hu, SL; Huang, ZH; Wang, P, A non-monotone smoothing Newton algorithm for solving nonlinear complementarity problems, Optim Methods Softw, 24, 447-460, (2009) · Zbl 1173.90552 |

[22] | Ni, T; Wang, P, A smoothing-type algorithm for solving nonlinear complementarity problems with a non-monotone line search, Appl Math Comput, 216, 2207-2214, (2010) · Zbl 1194.65080 |

[23] | Zhang, Y; Huang, ZH, A nonmonotone smoothing-type algorithm for solving a system of equalities and inequalities, J Comput Appl Math, 233, 2312-2321, (2010) · Zbl 1181.90265 |

[24] | Gowda, MS; Tawhid, MA, Existence and limiting behavior of trajectories associated with P\textsubscript{0}-equations, Comput Optim Appl, 12, 229-251, (1999) · Zbl 1040.90563 |

[25] | Gowda, MS; Sznajder, JJ, Weak univalence and connectedness of inverse images of continuous functions, Math Oper Res, 24, 255-261, (1999) · Zbl 0977.90060 |

[26] | Ravindran, G; Gowda, MS, Regularization of P\textsubscript{0}-functions in box variational inequality problems, SIAM J Optim, 11, 748-760, (2001) · Zbl 1010.90083 |

[27] | Qi, L; Sun, J, A nonsmooth version of newton’s method, Math Program, 58, 353-367, (1993) · Zbl 0780.90090 |

[28] | Fischer, A, Solution of monotone complementarity problems with Lipschitzian functions, Math Program, 76, 513-532, (1997) · Zbl 0871.90097 |

[29] | Qi, HD, A regularized smoothing Newton method for box constrained variational inequality problems with P0-functions, SIAM J Optim, 10, 315-330, (2000) |

[30] | Dirkse, S; Ferris, M, MCPLIB: a collection of nonlinear mixed complementarity problems, Optim Methods Softw, 5, 319-345, (1995) |

[31] | Andreani, R; Friedlander, A; Santos, S, On the resolution of the generalized nonlinear complementarity problem, SIAM J Optim, 12, 303-321, (2001) · Zbl 1006.65068 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.