Abstract
In this paper, building upon subgradient techniques and viscosity-type approximations, we propose a simple projection algorithm for solving the lexicographic Ky Fan inequality in a real Hilbert space, where the lower level is a variational inequality problem. By choosing suitable regularization parameters, a strong convergence of the proposed algorithm is established under mild assumptions imposed on the cost function. Some simple numerical examples are given to illustrate the performance of the proposed algorithm.
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Anh, P.N.: A new extragradient iteration algorithm for bilevel variational inequalities. Acta Math. Vietnam. 37, 95–107 (2012)
Anh, P.N.: An interior proximal method for solving pseudomonotone nonlipschitzian multivalued variational inequalities. Nonlinear Anal. Forum 14, 27–42 (2009)
Anh, P.N., Kim, J.K.: Outer approximation algorithms for pseudomonotone equilibrium problems. Comput. Math. Appl. 61, 2588–2595 (2011)
Anh, P.N., Kim, J.K., Muu, L.D.: An extragradient algorithm for solving bilevel pseudomonotone variational inequalities. J. Glob. Optim. 52, 627–639 (2012)
Anh, P.N., Muu, L.D., Strodiot, J.J.: Generalized projection method for non-Lipschitz multivalued monotone variational inequalities. Acta Math. Vietnam. 34, 67–79 (2009)
Anh, P.N., Muu, L.D., Hien, N.V., Strodiot, J.J.: Using the Banach contraction principle to implement the proximal point method for multivalued monotone variational inequalities. J. Optim. Theory Appl. 124, 285–306 (2005)
Baiocchi, C., Capelo, A.C.: Variational and Quasivariational Inequalities: Applications to Free Boundary Problems. Wiley, New York (1984)
Ceng, L.C., Cubiotti, P., Yao, J.C.: An implicit iterative scheme for monotone variational inequalities and fixed point problems. Nonlinear Anal. 69, 2445–2457 (2008)
Ceng, L.C., Yao, J.C.: Relaxed viscosity approximation methods for fixed point problems and variational inequality problems. Nonlinear Anal. 69, 3299–3309 (2008)
Daniele, P., Giannessi, F., Maugeri, A.: Equilibrium Problems and Variational Models. Kluwer Academic Publishers, Dordrecht (2003)
Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)
Fan, K.: A minimax inequality and applications. In: Shisha, O (ed.) Inequality III, pp 103–113. Academic Press, New York (1972)
Gao, X.-B., Liao, L.-Z., Qi, L.: A novel neural network for variational inequalities with linear and nonlinear constraints. IEEE Trans. Neural Netw. 16, 1305–1317 (2005)
Glackin, J., Ecker, J.G., Kupferschmid, M.: Solving bilevel linear programs using multiple objective linear programming. J. Optim. Theory Appl. 140, 197–212 (2009)
Iiduka, H.: Strong convergence for an iterative method for the triple-hierarchical constrained optimization problem. Nonlinear Anal. 71, e1292–e1297 (2009)
Kalashnikov, V.V., Kalashinikova, N.I.: Solving two-level variational inequality. J. Glob. Optim. 8, 289–294 (1996)
Konnov, I.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2001)
Lu, X.W., Xu, H.K., Yin, X.M.: Hybrid methods for a class of monotone variational inequalities. Nonlinear Anal. 71, 1032–1041 (2009)
Luo, Z.-Q., Pang, J.-S., Ralph, D.: Mathematical Programs with Equilibrum Constraints. Cambridge University Press, Cambridge (1996)
Maingé, P.-E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 16, 899–912 (2008)
Maingé, P.-E.: Projected subgradient techniques and viscosity methods for optimization with variational inequality constraints. Eur. J. Oper. Res. 205, 501–506 (2010)
Maingé, P.-E., Moudafi, A.: Strong convergence of an iterative method for hierarchical fixed-point problems. Pac. J. Optim. 3, 529–538 (2007)
Moudafi, A.: Proximal methods for a class of bilevel monotone equilibrium problems. J. Glob. Optim. 47, 287–292 (2010)
Sabharwal, A., Potter, L.C.: Convexly constrained linear inverse problems: iterative least-squares and regularization. IEEE Trans. Signal Process. 46, 2345–2352 (1998)
Solodov, M.: An explicit descent method for bilevel convex optimization. J. Convex Anal. 14, 227–237 (2007)
Quoc, T.D., Anh, P.N., Muu, L.D.: Dual extragradient algorithms extended to equilibrium problems. J. Glob. Optim. 52, 139–159 (2012)
Quoc, T.D., Muu, L.D., Hien, N.V: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008)
Trujillo-Cortez, R., Zlobec, S.: Bilevel convex programming models. Optimization 58, 1009–1028 (2009)
Xu, H.-K.: Viscosity method for hierarchical fixed point approach to variational inequalities. Taiwan. J. Math. 14, 463–478 (2010)
Xu, M.H., Li, M., Yang, C.C.: Neural networks for a class of bi-level variational inequalities. J. Glob. Optim. 44, 535–552 (2009)
Yao, Y., Marino, G., Muglia, L.: A modified Korpelevich’s method convergent to the minimum-norm solution of a variational inequality. Optimization 63, 559–569 (2014)
Acknowledgements
This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number “101.02-2017.15”.
We are very grateful to two anonymous referees for their really helpful and constructive comments that helped us very much in improving the paper.
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Anh, P.N., Thuy, L.Q. & Anh, T.T.H. Strong Convergence Theorem for the Lexicographic Ky Fan Inequality. Vietnam J. Math. 46, 517–530 (2018). https://doi.org/10.1007/s10013-017-0253-z
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DOI: https://doi.org/10.1007/s10013-017-0253-z