TWO STEP ALGORITHM FOR SOLVING REGULARIZED GENERALIZED MIXED VARIATIONAL INEQUALITY PROBLEM
1
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
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 Θ-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 Θ-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].
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 Θ-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 Θ-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].
참고문헌 (19)
- J. P. Aubin, "Differential Inclusions" (1984), Springer-Verlag
- E. Blum, "From optimization and variational inequalities to equilibrium problems", Math. Student, 63(1), pp.123~145 (1994)
- K. C. Border, "Fixed Point Theorems with Applications to Economics and Game Theory" (1985), Cambridge University Press
- M. Bounkhel, "Iterative schemes to solve nonconvex variational problems", JIPAM. J. Inequal. Pure Appl. Math., 4(1), pp.14~ (2003)
- F. H. Clarke, "Nonsmooth Analysis and Control Theory" (1998), Springer-Verlag
- F. Giannessi, "Variational Inequalities and Network Equilibrium Problems" (1995), Plenum Press
- F. Giannessi, "Equilibrium Problems: Non-smooth Optimization and Variational Inequality Models" (2001), Kluwer Academic Publishers
- R. Glowinski, "Numerical Analysis of Variational Inequalities" (1981), North-Holland Publishing Co.
- K. R. Kazmi, "Iterative approximation of solution of generalized mixed set-valued variational inequality problem", Math. Inequal. Appl., 10(3), pp.677~691 (2007)
- A. Moudafi, "An algorithmic approach to prox-regular variational inequalities", Appl. Math. Comput., 155(3), pp.845~852 (2004)
- S. B. Nadler, Jr., "Multi-valued contraction mappings", Pacific J. Math., 30, pp.475~488 (1969)
- M. A. Noor, "Auxiliary principle technique for equilibrium problems", J. Optim. Theory Appl., 122(2), pp.371~386 (2004)
- M. A. Noor, "Regularized mixed quasi equilibrium problems", J. Appl. Math. Comput., 23(1), pp.183~191 (2007)
- L.-P. Pang, "A modified predictor-corrector algorithm for solving nonconvex generalized variational inequality", Comput. Math. Appl., 54(3), pp.319~325 (2007)
- M. Patriksson, "Nonlinear Programming and Variational Inequality Problems" (1999), Kluwer Academic Publishers
- R. A. Poliquin, "Prox-regular functions in variational analysis", Trans. Amer. Math. Soc., 348(5), pp.1805~1838 (1996)
- R. T. Rockafellor, "Variational Analysis" (1998), Springer-Verlag
- G. Stampacchia, "Formes bilinéaires coercitives sur les ensembles convexes", C. R. Acad. Sci. Paris, 258, pp.4413~4416 (1964)
- F. Q. Xia, "Predictor-corrector algorithms for solving generalized mixedimplicit quasi-equilibrium problems", Appl. Math. Comput., 188(1), pp.173~179 (2007)