A projection-type method for generalized variational inequalities with dual solutions

Volume 10, Issue 9, pp 4812--4821
Publication Date: September 15, 2017 Submission Date: August 16, 2016
• 965 Views

Authors

Ming Zhu - School of Science, Guangxi University for Nationalities, Nanning, Guangxi 530006, P. R. China. Guo-Ji Tang - School of Science, Guangxi University for Nationalities, Nanning, Guangxi 530006, P. R. China.

Abstract

In this paper, a new projection-type method for generalized variational inequalities is introduced in Euclidean spaces. Under the assumption that the dual variational inequality has a solution, we show that the proposed method is well-defined and prove that the sequence generated by the proposed method is convergent to a solution, where the condition is strictly weaker than the pseudomonotonicity of the mapping used by some authors. We provide an example to support our results. Compared with the recent works of Li and He [F.-L. Li, Y.-R. He, J. Comput. Appl. Math., ${\bf 228}$ (2009), 212--218], and Fang and He [C.-J. Fang, Y.-R. He, Appl. Math. Comput., ${\bf 217}$ (2011), 9543--9551], condition (A3) is removed. Moreover, the results presented in this paper also generalize and improve some known results given in other literature.

Share and Cite

ISRP Style

Ming Zhu, Guo-Ji Tang, A projection-type method for generalized variational inequalities with dual solutions, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 9, 4812--4821

AMA Style

Zhu Ming, Tang Guo-Ji, A projection-type method for generalized variational inequalities with dual solutions. J. Nonlinear Sci. Appl. (2017); 10(9):4812--4821

Chicago/Turabian Style

Zhu, Ming, Tang, Guo-Ji. "A projection-type method for generalized variational inequalities with dual solutions." Journal of Nonlinear Sciences and Applications, 10, no. 9 (2017): 4812--4821

Keywords

• Projection method
• generalized variational inequality
• dual variational inequality.

•  90C33

References

• [1] J. P. Aubin, I. Ekeland , Applied nonlinear analysis , Pure and Applied Mathematics (New York), AWiley-Interscience Publication, John Wiley & Sons, Inc., New York (1984)

• [2] C.-J. Fang, Y.-R. He, A double projection algorithm for multi-valued variational inequalities and a unified framework of the method, Appl. Math. Comput., 217 (2011), 9543–9551.

• [3] N. Hadjisavvas, S. Schaible, On strong pseudomonotonicity and (semi)strict quasimonotonicity, J. Optim. Theory Appl., 79 (1993), 139–155.

• [4] N. Hadjisavvas, S. Schaible, Quasimonotone variational inequalities in Banach spaces, J. Optim. Theory Appl., 90 (1996), 95–111.

• [5] Y.-R. He , A new double projection algorithm for variational inequalities, J. Comput. Appl. Math., 185 (2006), 166–173.

• [6] A. N. Iusem, M. Nasri, Korpelevich’s method for variational inequality problems in Banach spaces, J. Global Optim., 50 (2011), 59–76.

• [7] A. N. Iusem, L. R. L. Pérez , An extragradient-type algorithm for non-smooth variational inequalities, Optimization, 48 (2000), 309–332.

• [8] A. N. Iusem, B. F. Svaiter , A variant of Korpelevich’s method for variational inequalities with a new search strategy, Optimization, 42 (1997), 309–321.

• [9] I. V. Konnov, On quasimonotone variational inequalities, J. Optim. Theory Appl., 99 (1998), 165–181.

• [10] G. M. Korpelevič , An extragradient method for finding saddle points and for other problems, (Russian) ´Ekonom. i Mat. Metody, 12 (1976), 747–756.

• [11] F.-L. Li, Y.-R. He , An algorithm for generalized variational inequality with pseudomonotone mapping , J. Comput. Appl. Math., 228 (2009), 212–218.

• [12] P. Marcotte, D. L. Zhu , A cutting plane method for solving quasimonotone variational inequalities, Comput. Optim. Appl., 20 (2001), 317–324.

• [13] M. V. Solodov, B. F. Svaiter, A new projection method for variational inequality problems, SIAM J. Control Optim., 37 (1999), 765–766.

• [14] G.-J. Tang, N.-J. Huang, Korpelevich’s method for variational inequality problems on Hadamard manifolds, J. Global Optim., 54 (2012), 493–509.

• [15] G.-J. Tang, X. Wang, H.-W. Liu , A projection-type method for variational inequalities on Hadamard manifolds and verification of solution existence, Optimization, 64 (2015), 1081–1096.

• [16] G.-J. Tang, M. Zhu, H.-W. Liu, A new extragradient-type method for mixed variational inequalities, Oper. Res. Lett., 43 (2015), 567–572.

• [17] M.-L. Ye, Y.-R. He, A double projection method for solving variational inequalities without monotonicity, Comput. Optim. Appl., 60 (2015), 141–150.