An Extragradient Method for Solving Variational Inequalities without Monotonicity

A new extragradient projection method, which does not require generalized monotonicity, is devised in this paper. In order to ensure its global convergence, we assume only that the Minty variational inequality has a solution. In particular, it applies to quasimonotone variational inequalities having a nontrivial solution.

[1]  Duong Viet Thong,et al.  Inertial extragradient algorithms for strongly pseudomonotone variational inequalities , 2018, J. Comput. Appl. Math..

[2]  Boris Polyak,et al.  Constrained minimization methods , 1966 .

[3]  A. Goldstein.  Convex programming in Hilbert space , 1964 .

[4]  Siegfried Schaible,et al.  Quasimonotone variational inequalities in Banach spaces , 1996 .

[5]  Jean Jacques Strodiot,et al.  A new class of hybrid extragradient algorithms for solving quasi-equilibrium problems , 2013, J. Glob. Optim..

[6]  Phan Tu Vuong,et al.  On the Weak Convergence of the Extragradient Method for Solving Pseudo-Monotone Variational Inequalities , 2018, Journal of Optimization Theory and Applications.

[7]  M. Solodov,et al.  A New Projection Method for Variational Inequality Problems , 1999 .

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

[9]  Yiran He.  A new double projection algorithm for variational inequalities , 2006 .

[10]  A. Iusem,et al.  A variant of korpelevich’s method for variational inequalities with a new search strategy , 1997 .

[11]  P. Tseng,et al.  Modified Projection-Type Methods for Monotone Variational Inequalities , 1996 .

[12]  J. Z. Zhang,et al.  Local Convergence Behavior of Some Projection-Type Methods for Affine Variational Inequalities , 2001 .

[13]  F. Giannessi.  On Minty Variational Principle , 1998 .

[14]  I. Konnov.  Combined Relaxation Methods for Variational Inequalities , 2000 .

[15]  Duong Viet Thong,et al.  New extragradient-like algorithms for strongly pseudomonotone variational inequalities , 2018, J. Glob. Optim..

[16]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[17]  I. Konnov.  Combined Relaxation Methods for Generalized Monotone Variational Inequalities , 2007 .

[18]  Yiran He,et al.  A double projection method for solving variational inequalities without monotonicity , 2014, Computational Optimization and Applications.

[19]  G. M. Korpelevich.  The extragradient method for finding saddle points and other problems , 1976 .

[20]  Naihua Xiu,et al.  Unified Framework of Extragradient-Type Methods for Pseudomonotone Variational Inequalities , 2001 .

[21]  R. Rockafellar.  Monotone Operators and the Proximal Point Algorithm , 1976 .

[22]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .