Abstract
We describe a fairly broad class of algorithms for solving variational inequalities, global convergence of which is based on the strategy of generating a hyperplane separating the current iterate from the solution set. The methods are shown to converge under very mild assumptions. Specifically, the problem mapping is only assumed to be continuous and pseudomonotone with respect to at least one solution. The strategy to obtain (super)linear rate of convergence is also discussed. The algorithms in this class differ in the tools which are used to construct the separating hyperplane. Our general scheme subsumes an extragradient-type projection method, a globally and locally super linearly convergent Josephy-Newton-type method, a certain minimization-based method, and a splitting technique.
This research is supported in part by CNPq Grant 300734/95-6 and by PRONEX-Optimization.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. A. Auslender. Optimisation Méthodes Numériques. Masson, Paris, 1976.
J. F. Bonnans. Local analysis of Newton-type methods for variational inequalities and nonlinear programming. Applied Mathematics and Optimization, 29:161–186, 1994.
R.W. Cottle, F. Giannessi, and J.-L. Lions. Variational Inequalities and Complementarity Problems: Theory and Applications. Wiley, New York, 1980.
J.-P. Crouzeix. Characterizations of generalized convexity and generalized monotonicity, A survey. In J.-P. Crouzeix et al., editor, Generalized convexity, generalized monotonicity: Recent results, pages 237–256. Kluwer Academic Publishers, 1998.
M. C. Ferris and J.-S. Pang (editors). Complementarity and variational problems: State of the Art. SIAM Publications, 1997.
M. C. Ferris and J.-S. Pang. Engineering and economic applications of complementarity problems. SIAM Review, 39:669–713, 1997.
M.C. Ferris and O.L. Mangasarian. Error bounds and strong upper semicontinuity for monotone affine variational inequalities. Annals of Operations Research, 47:293–305, 1993.
R. Glowinski, J.-L. Lions, and R. Trémolières. Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam, 1981.
P.T. Harker and J.-S. Pang. Finite-dimensional variational inequality problems: A survey of theory, algorithms and applications. Mathematical Programming, 48:161–220, 1990.
A.N. Iusem and B.F. Svaiter. A variant of Korpelevich’s method for variational inequalities with a new search strategy. Optimization, 42:309–321, 1997.
N.H. Josephy. Newton’s method for generalized equations. Technical Summary Report 1965, Mathematics Research Center, University of Wisconsin, Madison, Wisconsin, 1979.
S. Karamardian. Complementarity problems over cones with monotone and pseudomonotone maps. Journal of Optimization Theory and Applications, 18:445–455, 1976.
R. De Leone, O.L. Mangasarian, and T.-H. Shiau. Multi-sweep asynchronous parallel successive overrelaxation for the nonsymmet-ric linear complementarity problem. Annals of Operations Research, 22:43–54, 1990.
W. Li. Remarks on matrix splitting algorithms for symmetric linear complementarity problems. SIAM Journal on Optimization, 3:155–163, 1993.
Z.-Q. Luo and P. Tseng. Error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem. SIAM Journal on Optimization, 2:43–54, 1992.
O.L. Mangasarian. Convergence of iterates of an inexact matrix splitting algorithm for the symmetric monotone linear complementarity problem. SIAM Journal on Optimization, 1:114–122, 1991.
J.-S. Pang. Inexact Newton methods for the nonlinear complementarity problem. Mathematical Programming, 36:54–71, 1986.
S. M. Robinson. Strongly regular generalized equations. Mathematics of Operations Research, 5:43–62, 1980.
R.T. Rockafellar. Monotone operators and the proximal point algorithm. SIAM Journal on Control and Optimization, 14:877–898, 1976.
S. Schaible, S. Karamardian, and J.-P. Crouzeix. Characterizations of generalized monotone maps. Journal of Optimization Theory and Applications, 76:399–413, 1993.
M. V. Solodov and B. F. Svaiter. A truly globally convergent Newton-type method for the monotone nonlinear complementarity problem, 1998. SIAM Journal on Optimization 10 (2000), 605–625.
M. V. Solodov and B. F. Svaiter. A new projection method for variational inequality problems. SIAM Journal on Control and Optimization, 37:765–776, 1999.
M. V. Solodov and P. Tseng. Modified projection-type methods for monotone variational inequalities. SIAM Journal on Control and Optimization, 34:1814–1830, 1996.
M.V. Solodov and B.F. Svaiter. A hybrid approximate extragradient-proximal point algorithm using the enlargement of a maximal monotone operator, 1998. Set-Valued Analysis 7 (1999), 323–345.
M.V. Solodov and B.F. Svaiter. A globally convergent inexact Newton method for systems of monotone equations. In M. Fukushima and L. Qi, editors, Reformulation — Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, pages 355–369. Kluwer Academic Publishers, 1999.
M.V. Solodov and B.F. Svaiter. A hybrid projection — proximal point algorithm. Journal of Convex Analysis, 6:59–70, 1999.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Solodov, M.V. (2001). A Class of Globally Convergent Algorithms for Pseudomonotone Variational Inequalities. In: Ferris, M.C., Mangasarian, O.L., Pang, JS. (eds) Complementarity: Applications, Algorithms and Extensions. Applied Optimization, vol 50. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3279-5_14
Download citation
DOI: https://doi.org/10.1007/978-1-4757-3279-5_14
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-4847-2
Online ISBN: 978-1-4757-3279-5
eBook Packages: Springer Book Archive