A Descent-projection Method for Solving Monotone Structured Variational Inequalities
In this paper, a new descent-projection method with a
new search direction for monotone structured variational inequalities
is proposed. The method is simple, which needs only projections
and some function evaluations, so its computational load is very tiny.
Under mild conditions on the problem-s data, the method is proved to
converges globally. Some preliminary computational results are also
reported to illustrate the efficiency of the method.
[1] A.Nagurney, Network Ecomonics, A Variational Inequality Approach,
Kluwer Academics Publishers, Dordrect, 1993.
[2] F.Facchinei, J.S. Pang, Finite-Dimensional Variational Inequalities and
Complementarity Problems, vols. I and II, Springer Verlag, Berlin, 2003.
[3] M.K .Ng, F. Wang and X.M. Yuan, Inexact alternating direction methods
for image recovery. SIAM Journal on Scientific Computing, 2011, 33(4):
1643-1668.
[4] D. Gabay, Applications of the method of multipliers to variational inequalities.
In:M.Fortin and R.Glowinski(Eds) Augmented Lagrange Methods:
Applications to the Solution of Boundary valued Problems(Amsterdam:
North Holland),1983: 299-331.
[5] D. Gabay, and B. Mercier, A dual algorithm for the solution of nonlinear
variational problem svia finite-element approximations. Computers and
Mathematics with Applications, 1976, 2:17-40.
[6] Han, D.R. A modified alternating direction method for variational inequality
problems. Applied Mathematics and Optimization, 2002, 45: 63-74.
[7] Ye, C. H. and Yuan, X. M. A descent method for structured monotone
variational inequalities. Optimization Methods and Software, 2007, 22(2):
329-338.
[8] He B.S., Parallel splitting augmented Lagrangian methods for monotone
structured variational inequalities. Computational Optimization and Applications,
2009, 42(2): 195-212.
[9] X.H. Yan, D.R. Han, W.Y. Sun, A self-adaptive projection method with
improved step-size for solving variational inequalities. Computers and
Mathematics with Applications, 2008, 55(4): 819-832.
[10] X.Y. Zheng, H.W. Liu, Solving variational inequalities by a modified
projection method with an effective step-size. Applied Mathematics and
Computation, 2010, 216(12): 3778-3785.
[11] Z. Yu, H. Shao, G.D. Wang, Modified self-adaptive projection method
for solving pseudomonotone variational inequalities. Applied Mathematics
and Computation, 2011, 217(20): 8052-8060.
[12] M. Sun, A projection-type alternating direction method for structured
monotone variational inequality problems. Journal of Anhui University(
Natural Sciences), 2009, 33(2): 12-14.
[1] A.Nagurney, Network Ecomonics, A Variational Inequality Approach,
Kluwer Academics Publishers, Dordrect, 1993.
[2] F.Facchinei, J.S. Pang, Finite-Dimensional Variational Inequalities and
Complementarity Problems, vols. I and II, Springer Verlag, Berlin, 2003.
[3] M.K .Ng, F. Wang and X.M. Yuan, Inexact alternating direction methods
for image recovery. SIAM Journal on Scientific Computing, 2011, 33(4):
1643-1668.
[4] D. Gabay, Applications of the method of multipliers to variational inequalities.
In:M.Fortin and R.Glowinski(Eds) Augmented Lagrange Methods:
Applications to the Solution of Boundary valued Problems(Amsterdam:
North Holland),1983: 299-331.
[5] D. Gabay, and B. Mercier, A dual algorithm for the solution of nonlinear
variational problem svia finite-element approximations. Computers and
Mathematics with Applications, 1976, 2:17-40.
[6] Han, D.R. A modified alternating direction method for variational inequality
problems. Applied Mathematics and Optimization, 2002, 45: 63-74.
[7] Ye, C. H. and Yuan, X. M. A descent method for structured monotone
variational inequalities. Optimization Methods and Software, 2007, 22(2):
329-338.
[8] He B.S., Parallel splitting augmented Lagrangian methods for monotone
structured variational inequalities. Computational Optimization and Applications,
2009, 42(2): 195-212.
[9] X.H. Yan, D.R. Han, W.Y. Sun, A self-adaptive projection method with
improved step-size for solving variational inequalities. Computers and
Mathematics with Applications, 2008, 55(4): 819-832.
[10] X.Y. Zheng, H.W. Liu, Solving variational inequalities by a modified
projection method with an effective step-size. Applied Mathematics and
Computation, 2010, 216(12): 3778-3785.
[11] Z. Yu, H. Shao, G.D. Wang, Modified self-adaptive projection method
for solving pseudomonotone variational inequalities. Applied Mathematics
and Computation, 2011, 217(20): 8052-8060.
[12] M. Sun, A projection-type alternating direction method for structured
monotone variational inequality problems. Journal of Anhui University(
Natural Sciences), 2009, 33(2): 12-14.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:56080", author = "Min Sun and Zhenyu Liu", title = "A Descent-projection Method for Solving Monotone Structured Variational Inequalities", abstract = "In this paper, a new descent-projection method with a
new search direction for monotone structured variational inequalities
is proposed. The method is simple, which needs only projections
and some function evaluations, so its computational load is very tiny.
Under mild conditions on the problem-s data, the method is proved to
converges globally. Some preliminary computational results are also
reported to illustrate the efficiency of the method.", keywords = "variational inequalities, monotone function, global
convergence.", volume = "5", number = "12", pages = "1940-5", }