TWO STEP ALGORITHM FOR SOLVING REGULARIZED GENERALIZED MIXED VARIATIONAL INEQUALITY PROBLEM

Title & Authors
TWO STEP ALGORITHM FOR SOLVING REGULARIZED GENERALIZED MIXED VARIATIONAL INEQUALITY PROBLEM

Abstract
In this paper, we consider a new class of regularized (nonconvex) generalized mixed variational inequality problems in real Hilbert space. We give the concepts of partially relaxed strongly mixed monotone and partially relaxed strongly $\small{\theta}$-pseudomonotone mappings, which are extension of the concepts given by Xia and Ding [19], Noor [13] and Kazmi et al. [9]. Further we use the auxiliary principle technique to suggest a two-step iterative algorithm for solving regularized (nonconvex) generalized mixed variational inequality problem. We prove that the convergence of the iterative algorithm requires only the continuity, partially relaxed strongly mixed monotonicity and partially relaxed strongly $\small{\theta}$-pseudomonotonicity. The theorems presented in this paper represent improvement and generalization of the previously known results for solving equilibrium problems and variational inequality problems involving the nonconvex (convex) sets, see for example Noor [13], Pang et al. [14], and Xia and Ding [19].
Keywords
regularized generalized mixed variational inequality problem;auxiliary problem;two-step iterative algorithm;convergence analysis;partially relaxed strongly mixed monotone mapping;regularized partially relaxed strongly $\small{\theta}$-pseudomonotone mapping;skew-symmetric function;
Language
English
Cited by
1.
Existence Results for Vector Mixed Quasi-Complementarity Problems, Journal of Mathematics, 2013, 2013, 1
2.
Implicit iterative method for approximating a common solution of split equilibrium problem and fixed point problem for a nonexpansive semigroup, Arab Journal of Mathematical Sciences, 2014, 20, 1, 57
References
1.
J. P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, New York, 1984.

2.
E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student 63 (1994), no. 1-4, 123–145.

3.
K. C. Border, Fixed Point Theorems with Applications to Economics and Game Theory, Cambridge University Press, Cambridge, 1985.

4.
M. Bounkhel, L. Tadj, and A. Hamdi, Iterative schemes to solve nonconvex variational problems, JIPAM. J. Inequal. Pure Appl. Math. 4 (2003), no. 1, Article 14, 14 pp.

5.
F. H. Clarke, Y. S. Ledyaev, R. J. Stern, and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, 178. Springer-Verlag, New York, 1998.

6.
F. Giannessi and A. Maugeri, Variational Inequalities and Network Equilibrium Problems, Plenum Press, New York, 1995.

7.
F. Giannessi, A. Maugeri, and P. M. Pardalos, Equilibrium Problems: Non-smooth Optimization and Variational Inequality Models, Kluwer Academic Publishers, Dordrecht, 2001.

8.
R. Glowinski, J. L. Lions, and R. Tremolieres, Numerical Analysis of Variational Inequalities, North-Holland Publishing Co., Amsterdam-New York, 1981.

9.
K. R. Kazmi, A. Khaliq, and A. Raouf, Iterative approximation of solution of generalized mixed set-valued variational inequality problem, Math. Inequal. Appl. 10 (2007), no. 3, 677–691.

10.
A. Moudafi, An algorithmic approach to prox-regular variational inequalities, Appl. Math. Comput. 155 (2004), no. 3, 845–852.

11.
S. B. Nadler, Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475–488.

12.
M. A. Noor, Auxiliary principle technique for equilibrium problems, J. Optim. Theory Appl. 122 (2004), no. 2, 371–386.

13.
M. A. Noor, Regularized mixed quasi equilibrium problems, J. Appl. Math. Comput. 23 (2007), no. 1-2, 183–191.

14.
L.-P. Pang, J. Shen, and H.-S. Song, A modified predictor-corrector algorithm for solving nonconvex generalized variational inequality, Comput. Math. Appl. 54 (2007), no. 3, 319–325.

15.
M. Patriksson, Nonlinear Programming and Variational Inequality Problems, Kluwer Academic Publishers, Dordrecht, 1999.

16.
R. A. Poliquin and R. T. Rockafellar, Prox-regular functions in variational analysis, Trans. Amer. Math. Soc. 348 (1996), no. 5, 1805–1838.

17.
R. T. Rockafellor and R. J.-B.Wets, Variational Analysis, Springer-Verlag, Berlin, 1998.

18.
G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris 258 (1964), 4413–4416.

19.
F. Q. Xia and X. P. Ding, Predictor-corrector algorithms for solving generalized mixed implicit quasi-equilibrium problems, Appl. Math. Comput. 188 (2007), no. 1, 173–179.