Comparison of Two Kinds of Prediction-Correction Methods for Monotone Variational Inequalities

In this paper, we study the relationship between the forward-backward splitting method and the extra-gradient method for monotone variational inequalities. Both of the methods can be viewed as prediction-correction methods. The only difference is that they use different search directions in the correction-step. Our analysis explains theoretically why the extra-gradient methods usually outperform the forward-backward splitting methods. We suggest some modifications for the two methods and numerical results are given to verify the superiority of the modified methods.

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

[2]  L. Liao,et al.  Improvements of Some Projection Methods for Monotone Nonlinear Variational Inequalities , 2002 .

[3]  Bingsheng He,et al.  Inexact implicit methods for monotone general variational inequalities , 1999, Math. Program..

[4]  H. H. Rachford,et al.  On the numerical solution of heat conduction problems in two and three space variables , 1956 .

[5]  Patrick T. Harker,et al.  Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications , 1990, Math. Program..

[6]  Yinyu Ye,et al.  An Efficient Algorithm for Minimizing a Sum of Euclidean Norms with Applications , 1997, SIAM J. Optim..

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

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

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

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

[11]  R. Glowinski,et al.  Numerical Methods for Nonlinear Variational Problems , 1985 .

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

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

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

[15]  Masao Fukushima,et al.  A globally convergent Newton method for solving strongly monotone variational inequalities , 1993, Math. Program..

[16]  P. Marcotte.  APPLICATION OF KHOBOTOVS ALGORITHM TO VARIATIONAL INEQUALITIES ANT) NETWORK EQUILIBRIUM PROBLEMS , 1991 .

[17]  Richard W. Cottle,et al.  Linear Complementarity Problem. , 1992 .

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

[19]  B. Curtis Eaves,et al.  On the basic theorem of complementarity , 1971, Math. Program..

[20]  Bin-Xin He.  Solving a class of linear projection equations , 1994 .

[21]  P. Lions,et al.  Splitting Algorithms for the Sum of Two Nonlinear Operators , 1979 .

[22]  B. He.  A class of projection and contraction methods for monotone variational inequalities , 1997 .

[23]  Defeng Sun,et al.  A class of iterative methods for solving nonlinear projection equations , 1996 .

[24]  P. Marcotte,et al.  A note on a globally convergent Newton method for solving monotone variational inequalities , 1986 .

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

[26]  R. Tyrrell Rockafellar,et al.  Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming , 1976, Math. Oper. Res..