Abstract
In this paper, using the approximate duality mapping, we introduce the definition of weak sharpness of the solution set to a mixed variational inequality in Banach spaces. In terms of the primal gap function associated to the mixed variational inequality, we give several characterizations of the weak sharpness.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Auslender, A.: Optimisation. M\(\acute{\text{ e }}\)thodes num\(\acute{\text{ e }}\)riques. Ma\(\hat{{\i }}\)trise de Math\(\acute{\text{ e }}\)matiques et Applications Fondamentales. Masson, Paris (1976)
Burke, J.V., Ferris, M.C.: Weak sharp minima in mathematical programming. SIAM J. Control. Optim. 31(5), 1340–1359 (1993)
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983). [Reprinted as Classics Appl. Math. 5, SIAM, Philadelphia (1990)]
Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I and II. Springer, New York (2003)
Hu, Y.H., Song, W.: Weak sharp solutions for variational inequalities in Banach spaces. J. Math. Anal. Appl. 374(1), 118–132 (2011)
Liu, Y.N., Wu, Z.L.: Characterization of weakly sharp solutions of a variational inequality by its primal gap function. Optim. Lett. 10(3), 563–576 (2016)
Marcotte, P., Zhu, D.L.: Weak sharp solutions of variational inequalities. SIAM J. Optim. 9(1), 179–189 (1998)
Patriksson, M.: A Unified Framework of Descent Algorithms for Nonlinear Programs and Variational Inequalities, Ph.D. thesis, Department of Mathematics, Link\(\ddot{\text{ o }}\)ping Institute of Technology, Link\(\ddot{\text{ o }}\)ping, Sweden (1993)
Pappalardo, M., Mastroeni, G., Passacantando, M.: Merit functions: a bridge between optimization and equilibria. 4OR 12(1), 1–33 (2014)
Salmon, G., Strodiot, J.J., Nguyen, V.H.: A bundle method for solving variational inequalities. SIAM J. Optim. 14(3), 869–893 (2003)
Wu, Z.L., Wu, S.Y.: Weak sharp solutions of variational inequalities in Hilbert spaces. SIAM J. Optim. 14(4), 1011–1027 (2004)
Zhang, J.Z., Wan, C.Y., Xiu, N.H.: The dual gap functions for variational inequalities. Appl. Math. Optim. 48(2), 129–148 (2003)
Zheng, X.Y., Ng, K.F.: Linear regularity for a collection of subsmooth sets in Banach spaces. SIAM J. Optim. 19(1), 62–76 (2008)
Zhong, R.Y., Huang, N.J.: Stability analysis for Minty mixed variational inequality in reflexive Banach spaces. J. Optim. Theory Appl. 147(3), 454–472 (2010)
Acknowledgements
The authors express their deep gratitude to the referees for helpful comments which improve this paper. The research was supported by the National Natural Science Foundation of China under Grant Nos. 11461080, 11371312 and 11261067.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Huang, H., He, M. Weak sharp solutions of mixed variational inequalities in Banach spaces. Optim Lett 12, 287–299 (2018). https://doi.org/10.1007/s11590-017-1112-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-017-1112-2