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Proximal point algorithms for general variational inequalities. (English) Zbl 1198.90374
This paper studies the application of the classical proximal point algorithms (PPAs) to general variational inequalities (GVIs), which have wide applications in various fields. Some existing PPAs for classical variational inequalities, including both the exact and inexact versions, are extended to solving GVIs. Consequently, a unified framework of PPA-based algorithms is provided. New algorithms proposed by the authors include some existing methods as special cases.

90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
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[1] Giannessi, F., Maugeri, A., Pardalos, P.M. (eds.): Equilibrium Problems and Variational Models. Kluwer Academic, Dordrecht (2001) · Zbl 0979.00025
[2] Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Series in Operations Research. Springer, Berlin (2003) · Zbl 1062.90002
[3] He, B.S.: A Goldstein’s type projection method for a class of variant variational inequalities. J. Comput. Math. 17, 425–434 (1999) · Zbl 0936.65078
[4] Li, M., Yuan, X.M.: An improved Goldstein’s type method for a class of variant variational inequalities. J. Comput. Appl. Math. 214(1), 304–312 (2008) · Zbl 1162.65038
[5] Giannessi, F., Maugeri, A. (eds.): Variational Analysis and Applications. Springer, New York (2005) · Zbl 1077.49001
[6] 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(222), 519–540 (1998) · Zbl 0894.90143
[7] Noor, M.A.: General variational inequalities. Appl. Math. Lett. 1, 119–121 (1988) · Zbl 0655.49005
[8] Pang, J.S., Yao, J.C.: On a generalization of a normal map and equation. SIAM J. Control Optim. 33, 168–184 (1995) · Zbl 0827.90131
[9] Xiu, N.H., Zhang, J.Z.: Global projection-type error bounds for general variational inequalities. J. Optim. Theory Appl. 112(1), 213–228 (2002) · Zbl 1005.49004
[10] Vandenberghe, L., De Moor, B.L., Vandewalle, J.: The generalized linear complementary problem applied to the complete analysis of resistive piecewise-linear circuits. IEEE Trans. Circuits Syst. II 36, 1382–1391 (1989)
[11] Li, M., Liao, L.-Z., Yuan, X.M.: The variant VI model for second-best road pricing. Manuscript (2007)
[12] He, B.S.: Inexact implicit methods for monotone general variational inequalities. Math. Program. 86, 199–217 (1999) · Zbl 0979.49006
[13] Noor, M.A., Wang, Y.J., Xiu, N.H.: Projection iterative schemes for general variational inequalities. J. Inequal. Pure Appl. Math. 3(3), 34 (2002) · Zbl 1142.49304
[14] Xia, Y.S., Wang, J.: A general projection neural network for solving monotone variational inequalities and related optimization problems. IEEE Trans. Neural Netw. 15(2), 318–328 (2004)
[15] Martinet, B.: Regularization d’inequations variationelles par approximations sucessives. Rev. Fr. Inf. Rech. Opér. 4, 154–159 (1970)
[16] Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976) · Zbl 0358.90053
[17] Rockafellar, R.T.: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1, 97–116 (1976) · Zbl 0402.90076
[18] Eaves, B.C.: On the basic theorem of complementarity. Math. Program. 1, 68–75 (1971) · Zbl 0227.90044
[19] Burachik, R.S., Iusem, A.N.: A generalized proximal point algorithm for the variational inequality problem in a Hilbert space. SIAM J. Optim. 8, 197–216 (1998) · Zbl 0911.90273
[20] Chen, G., Teboulle, M.: A proximal-based decomposition method for convex minimization problem. Math. Program. 64, 81–101 (1994) · Zbl 0823.90097
[21] Eckstein, J.: Approximate iterations in Bregman-function-based proximal algorithms. Math. Program. 83, 113–123 (1998) · Zbl 0920.90117
[22] Han, D.R., He, B.S.: A new accuracy criterion for approximate proximal point algorithms. J. Math. Anal. Appl. 263, 343–353 (2001) · Zbl 0995.65062
[23] He, B.S., Yang, Z.H., Yuan, X.M.: An approximate proximal-extragradient type method for monotone variational inequalities. J. Math. Anal. Appl. 300, 362–374 (2004) · Zbl 1068.65087
[24] Solodov, M.V., Svaiter, B.F.: A unified framework for some inexact proximal point algorithms. Numer. Funct. Anal. Optim. 22, 1013–1035 (2001) · Zbl 1052.49013
[25] Teboulle, M.: Convergence of proximal-like algorithms. SIAM J. Optim. 7, 1069–1083 (1997) · Zbl 0890.90151
[26] Auslender, A., Haddou, M.: An interior proximal point method for convex linearly constrained problems and its extension to variational inequalities. Math. Program. 71, 77–100 (1995) · Zbl 0855.90095
[27] Han, D.: A new hybrid generalized proximal point algorithm for variational inequality problems. J. Glob. Optim. 26(2), 125–140 (2003) · Zbl 1044.65055
[28] Auslender, A., Teboulle, M., Ben-Tiba, S.: A logarithmic-quadratic proximal method for variational inequalities. Comput. Optim. Appl. 12, 31–40 (1999) · Zbl 1039.90529
[29] Yuan, X.M.: The prediction-correction approach to nonlinear complementarity problems. Eur. J. Oper. Res. 176, 1357–1370 (2007) · Zbl 1102.90064
[30] Bertsekas, D.P., Tsitsiklis, J.N.: Parallel and Distributed Computation, Numerical Methods. Prentice-Hall, Englewood Cliffs (1989) · Zbl 0743.65107
[31] He, B.S., Yuan, X.M., Zhang, J.J.Z.: Comparison of two kinds of prediction-correction methods for monotone variational inequalities. Comput. Optim. Appl. 27(3), 247–267 (2004) · Zbl 1061.90111
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