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A nonmonotone smoothing Newton algorithm for solving general box constrained variational inequalities. (English) Zbl 1402.90193
Summary: 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
Software:
MCPLIB
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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
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