Solving Strongly Monotone Variational and Quasi-Variational Inequalities

In this paper we develop a new and efficient method for variational inequality with Lipschitz continuous strongly monotone operator. Our analysis is based on a new strongly convex merit function. We apply a variant of the developed scheme for solving quasivariational inequality. As a result, we significantly improve the standard sufficient condition for existence and uniqueness of their solutions. Moreover, we get a new numerical scheme, which rate of convergence is much higher than that of the straightforward gradient method.

[1]  U. Mosco.  Implicit variational problems and quasi variational inequalities , 1976 .

[2]  J. S. Pang,et al.  The Generalized Quasi-Variational Inequality Problem , 1982, Math. Oper. Res..

[3]  P. Bovy,et al.  Quasi-variational inequality formulation of the multiclass dynamic traffic assignment problem ☆ , 2003 .

[4]  Jaroslav Haslinger,et al.  The reciprocal variational approach to the Signorini problem with friction. Approximation results , 1984, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[5]  Muhammad Aslam Noor,et al.  ALGORITHMS FOR GENERAL MIXED QUASI VARIATIONAL INEQUALITIES , 2002 .

[6]  Muhammad Aslam Noor,et al.  On a class of quasi variational inequalities , 1996, Approximation Theory and its Applications.

[7]  Yurii Nesterov,et al.  Smooth minimization of non-smooth functions , 2005, Math. Program..

[8]  Yurii Nesterov,et al.  Dual extrapolation and its applications to solving variational inequalities and related problems , 2003, Math. Program..

[9]  F. Giannessi,et al.  Variational Analysis and Applications , 2005 .

[10]  P. Harker.  Generalized Nash games and quasi-variational inequalities , 1991 .

[11]  Yves Smeers,et al.  Spatial Oligopolistic Electricity Models with Cournot Generators and Regulated Transmission Prices , 1999, Oper. Res..

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

[13]  C. Baiocchi,et al.  Variational and quasivariational inequalities: Applications to free boundary problems , 1983 .

[14]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[15]  Jirí V. Outrata,et al.  A numerical approach to optimization problems with variational inequality constraints , 1995, Math. Program..

[16]  W. Oettli,et al.  On general nonlinear complementarity problems and quasi-equilibria , 1995 .

[17]  Laura Scrimali,et al.  The financial equilibrium problem with implicit budget constraints , 2008, Central Eur. J. Oper. Res..

[18]  Jaroslav Haslinger,et al.  Approximation of the signorini problem with friction, obeying the coulomb law , 1983 .

[19]  Yurii Nesterov,et al.  Introductory Lectures on Convex Optimization - A Basic Course , 2014, Applied Optimization.

[20]  Michal Kočvara,et al.  On a class of quasi-variational inequalities , 1995 .

[21]  Arkadi Nemirovski,et al.  Prox-Method with Rate of Convergence O(1/t) for Variational Inequalities with Lipschitz Continuous Monotone Operators and Smooth Convex-Concave Saddle Point Problems , 2004, SIAM J. Optim..

[22]  A. Bensoussan,et al.  Contrôle impulsionnel et inéquations quasi variationnelles , 1982 .

[23]  Jen-Chih Yao,et al.  The generalized quasi-variational inequality problem with applications , 1991 .

[24]  Masao Fukushima,et al.  Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems , 1992, Math. Program..

[25]  Hanif Salahuddin.  Projection Methods for Quasi-Variational Inequalities , 2004 .

[26]  John Darzentas,et al.  Problem Complexity and Method Efficiency in Optimization , 1983 .

[27]  A. Bensoussan.  Points de Nash Dans le Cas de Fonctionnelles Quadratiques et Jeux Differentiels lineaires a N Personnes , 1974 .

[28]  G. Minty.  On the generalization of a direct method of the calculus of variations , 1967 .