A double projection method for solving variational inequalities without monotonicity

We present a double projection algorithm for solving variational inequalities without monotonicity. If the solution of dual variational inequality does exist, then the sequence produced by our method is globally convergent to a solution. Under the same assumption, the sequence produced by known methods has only a subsequence converging to a solution. Numerical experiments are reported.

[1]  N. Xiu,et al.  Some recent advances in projection-type methods for variational inequalities , 2003 .

[2]  H. Kunzi,et al.  Lectu re Notes in Economics and Mathematical Systems , 1975 .

[3]  E. Khobotov.  Modification of the extra-gradient method for solving variational inequalities and certain optimization problems , 1989 .

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

[5]  Patrice Marcotte,et al.  A Cutting Plane Method for Solving Quasimonotone Variational Inequalities , 2001, Comput. Optim. Appl..

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

[7]  João X. da Cruz Neto,et al.  Interior Proximal Algorithm for Quasiconvex Programming Problems and Variational Inequalities with Linear Constraints , 2012, J. Optim. Theory Appl..

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

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

[10]  Alexander M. Rubinov,et al.  Generalized convexity and related topics , 2006 .

[11]  Wenyu Sun,et al.  A self-adaptive projection method with improved step-size for solving variational inequalities , 2008, Comput. Math. Appl..

[12]  E. Polak,et al.  Constrained Minimization Problems in Finite-Dimensional Spaces , 1966 .

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

[14]  Sun.  A NEW STEP-SIZE SKILL FOR SOLVING A CLASS OF NONLINEAR PROJECTION EQUATIONS , 1995 .

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

[16]  Nils Langenberg.  An Interior Proximal Method for a Class of Quasimonotone Variational Inequalities , 2012, J. Optim. Theory Appl..

[17]  Zhiwei Wang,et al.  A simple self-adaptive alternating direction method for linear variational inequality problems , 2007, Comput. Math. Appl..

[18]  S. Karamardian.  Complementarity problems over cones with monotone and pseudomonotone maps , 1976 .

[19]  H. Lo,et al.  Two new self-adaptive projection methods for variational inequality problems , 2002 .

[20]  Lamberto Cesari,et al.  Optimization-Theory And Applications , 1983 .

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

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

[23]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

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

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

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