On the Iteration Complexity of Some Projection Methods for Monotone Linear Variational Inequalities

Projection-type methods are important for solving monotone linear variational inequalities. In this paper, we analyze the iteration complexity of two projection methods and accordingly establish their worst-case sublinear convergence rates measured by the iteration complexity in both the ergodic and nonergodic senses. Our analysis does not require any error bound condition or the boundedness of the feasible set, and it is scalable to other methods of the same kind.

[1]  Y. Nesterov.  Gradient methods for minimizing composite objective function , 2007 .

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

[3]  W. Oettli,et al.  Mathematische Optimierung : Grundlagen und Verfahren , 1975 .

[4]  Bingsheng He,et al.  A new method for a class of linear variational inequalities , 1994, Math. Program..

[5]  Yurii Nesterov,et al.  Gradient methods for minimizing composite functions , 2012, Mathematical Programming.

[6]  Paul Tseng,et al.  A Modified Forward-backward Splitting Method for Maximal Monotone Mappings 1 , 1998 .

[7]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

[8]  G. Stampacchia,et al.  On some non-linear elliptic differential-functional equations , 1966 .

[9]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

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

[11]  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..

[12]  B. He.  Solving a Class of Linear Projection Equations ? , 1994 .

[13]  B. He,et al.  A MODIFIED PROJECTION AND CONTRACTION METHOD FOR A CLASS OF LINEAR COMPLEMENTARITY PROBLEMS , 2006 .

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

[15]  Bingsheng He,et al.  A projection and contraction method for a class of linear complementarity problems and its application in convex quadratic programming , 1992 .

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

[17]  Y. Nesterov.  A method for solving the convex programming problem with convergence rate O(1/k^2) , 1983 .