Abstract
In this paper, we present an alternating direction method for structured general variational inequalities. This method only needs functional values for given variables in the solution process and does not require the estimate of the co-coercive modulus. All the computing process are easily implemented and the global convergence is also presented under mild assumptions. Some preliminary computational results are given.
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Bertsekas, D.P., Tsitsiklis, J.N.: Parallel and Distributed Computation, Numerical Methods. Prentice-Hall, Englewood Cliffs (1989)
Eaves, B.C.: Computing stationary points. Math. Program. Stud. 7, 1–14 (1978)
Gabay, D.: Applications of the method of multipliers to variational inequalities. In: Fortin, M., Glowinski, R. (eds.) Augmented Lagrangian Methods: Applications to the Solution of Boundary-Valued Problems, pp. 299–331. North-Holland, Amsterdam (1983)
Gabay, D., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite-element approximations. Comput. Math. Appl. 2(1), 17–40 (1976)
Han, D.R.: A modified alternating direction method for variational inequality problems. Appl. Math. Optim. 45, 63–74 (2002)
Han, D.R., Lo, H.K.: New alternating direction method for a class of nonlinear variational inequality problems. J. Optim. Theory Appl. 112, 549–560 (2002)
Han, D.R., Lo, H.K.: A new stepsize rule in He and Zhou’s alternating direction method. Appl. Math. Lett. 15, 181–185 (2002)
He, B.S., Zhou, J.: A modified alternating direction method for convex minimization problems. Appl. Math. Lett. 13, 122–130 (2000)
He, B.S., He, X.Z., Liu, H.X., Wu, T.: Self-adaptive projection method for co-coercive variational inequalities. Eur. J. Oper. Res. 196, 43–48 (2009)
Jiang, Z.K., Bnouhachem, A.: A projection-based prediction-correction method for structured monotone variational inequalities. Appl. Math. Comput. 202, 747–759 (2008)
Li, M., Liao, L.Z., Yuan, X.M.: A modified descent projection method for co-coercive variational inequalities. Eur. J. Oper. Res. 189(2), 310–323 (2008)
Yang, J., Zang, Y.: Alternating direction algorithms for L1 problems in compressive sensing. SIAM J. Sci. Comput. 33(1), 250–278 (2011)
Zarantonello, E.H.: Projections on convex sets in Hilbert space and spectral theory. In: Zarantonello, H. (ed.) Contributions to Nonlinear Functional Analysis. Academic Press, New York (1971)
Zhang, W., Han, D.R.: A new alternating direction method for co-coercive variational inequality problems. Comput. Math. Appl. 57(7), 1168–1178 (2009)
Zhu, T., Yu, Z.G.: A simple proof for some important properties of the projection mapping. Math. Inequal. Appl. 7, 453–456 (2004)
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Bnouhachem, A., Noor, M.A., Khalfaoui, M. et al. An alternating direction method for general variational inequalities. J. Appl. Math. Comput. 38, 535–549 (2012). https://doi.org/10.1007/s12190-011-0495-y
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DOI: https://doi.org/10.1007/s12190-011-0495-y