Iterative methods for solving general quasi-variational inequalities

In this paper, we introduce a new class of variational inequalities, which is called the general quasi-variational inequality. We establish the equivalence among the general quasi variational inequality and implicit fixed point problems and the Wiener–Hopf equations. We use this equivalent formulation to discuss the existence of a solution of the general quasi-variational inequality. This equivalent formulation is used to suggest and analyze some iterative algorithms for solving the general quasi-variational inequality. We also discuss the convergence analysis of these iterative methods. Several special cases are also discussed.

[1]  Anna Nagurney,et al.  Variational Inequalities , 2009, Encyclopedia of Optimization.

[2]  Muhammad Aslam Noor,et al.  Some developments in general variational inequalities , 2004, Appl. Math. Comput..

[3]  Muhammad Aslam Noor,et al.  Sensitivity Analysis for Quasi-Variational Inequalities , 1997 .

[4]  Muhammad Aslam Noor.  Projection iterative methods for extended general variational inequalities , 2010 .

[5]  Muhammad Aslam Noor,et al.  EXISTENCE RESULTS FOR QUASI VARIATIONAL INEQUALITIES , 2007 .

[6]  Muhammad Aslam Noor.  SOME CLASSES OF VARIATIONAL INEQUALITIES , 1991 .

[7]  Muhammad Aslam Noor,et al.  General Wiener-Hopf equation technique for nonexpansive mappings and general variational inequalities in Hilbert spaces , 2008, Appl. Math. Comput..

[8]  Muhammad Aslam Noor,et al.  Extended general variational inequalities , 2009, Appl. Math. Lett..

[9]  Biao Qu,et al.  Merit functions for general mixed quasi-variational inequalities , 2010 .

[10]  R. Glowinski,et al.  Numerical Analysis of Variational Inequalities , 1981 .

[11]  M. Noor,et al.  Some aspects of variational inequalities , 1993 .

[12]  Muhammad Aslam Noor,et al.  Quasi variational inequalities , 1988 .

[13]  Peter Shi,et al.  Equivalence of variational inequalities with Wiener-Hopf equations , 1991 .

[14]  M. Noor.  General variational inequalities , 1988 .

[15]  Youzhong Guo.  ON VARIATIONAL INEQUALITIES , 1984 .

[16]  Muhammad Aslam Noor,et al.  An Iterative Scheme for a Class of Quasi Variational Inequalities , 1985 .

[17]  Muhammad Aslam Noor,et al.  Differentiable non-convex functions and general variational inequalities , 2008, Appl. Math. Comput..

[18]  Panos M. Pardalos,et al.  Nonlinear Analysis and Variational Problems , 2010 .

[19]  Panos M. Pardalos,et al.  From Convexity to Nonconvexity , 2011 .

[20]  Muhammad Aslam Noor,et al.  Wiener-hopf equations and variational inequalities , 1993 .

[21]  Muhammad Aslam Noor,et al.  Generalized Multivalued Quasi-Variational Inequalities , 1998 .

[22]  Pekka Neittaanmäki,et al.  Variational and Quasi-Variational Inequalities in Mechanics , 2007 .

[23]  M. Noor.  New approximation schemes for general variational inequalities , 2000 .

[24]  W. Hager.  Review: R. Glowinski, J. L. Lions and R. Trémolières, Numerical analysis of variational inequalities , 1983 .

[25]  Muhammad Aslam Noor.  ON MERIT FUNCTIONS FOR QUASIVARIATIONAL INEQUALITIES , 2007 .

[26]  Ralph Tyrell Rockafellar,et al.  Part I. Basic Concepts , 1970 .

[27]  P. Pardalos,et al.  Equilibrium problems : nonsmooth optimization and variational inequality models , 2004 .

[28]  M. Noor.  Some algorithms for general monotone mixed variational inequalities , 1999 .

[29]  Alain Bensoussan,et al.  Applications des Inequations Varia-tionnelles en Controle Stochastique , 1978 .