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Augmented Lagrangian methods for variational inequality problems

Published online by Cambridge University Press:  08 February 2010

Alfredo N. Iusem  and
Mostafa Nasri
Alfredo N. Iusem
Affiliation:
Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, RJ, CEP 22460-320, Brazil; [email protected]; [email protected]
Mostafa Nasri
Affiliation:
Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, RJ, CEP 22460-320, Brazil; [email protected]; [email protected]
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Abstract

We introduce augmented Lagrangian methods for solving finite dimensional variational inequality problemswhose feasible sets are defined by convex inequalities, generalizing the proximal augmented Lagrangian methodfor constrained optimization. At each iteration, primal variables are updated by solvingan unconstrained variational inequality problem, and then dual variables are updated through a closed formula.A full convergence analysis is provided, allowing for inexact solution of the subproblems.

Keywords

Augmented Lagrangian methodequilibrium probleminexact solutionproximal point methodvariational inequality problem
Type
Research Article
Information
RAIRO - Operations Research , Volume 44 , Issue 1 , January 2010 , pp. 5 - 25
Copyright
© EDP Sciences, ROADEF, SMAI, 2010

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References

Antipin, A.S., Equilibrium programming: proximal methods. Comput. Math. Math. Phys. 37 (1997) 1285–1296.
Antipin, A.S., Vasilev, F.P. and Stukalov, A.S., A regularized Newton method for solving equilibrium programming problems with an inexactly specified set. Comput. Math. Math. Phys. 47 (2007) 19–31. CrossRef
Auslender, A. and Teboulle, M., Lagrangian duality and related multiplier methods for variational inequality problems. SIAM J. Optim. 10 (2000) 1097–1115. CrossRef
Bertsekas, D.P., On penalty and multiplier methods for constrained optimization problems. SIAM J. Control Optim. 14 (1976) 216–235. CrossRef
Bianchi, M. and Pini, R., Coercivity conditions for equilibrium problems. J. Optim. Theory Appl. 124 (2005) 79–92. CrossRef
Bianchi, M. and Schaible, S., Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl. 90 (1996) 31–43. CrossRef
Blum, E. and Oettli, W., From optimization and variational inequalities to equilibrium problems. The Mathematics Student 63 (1994) 123–145.
Brezis, H., Nirenberg, L. and Stampacchia, S., A remark on Ky Fan minimax principle. Bolletino della Unione Matematica Italiana 6 (1972) 293–300.
J.D. Buys, Dual algorithms for constrained optimization problems, Ph.D. thesis, University of Leiden, The Netherlands (1972).
F. Facchinei and J.S. Pang, Finite-dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin (2003).
Ferris, M.C. and Pang, J.S., Engineering and economic applications of complementarity problems. SIAM Rev. 39 (1997) 669–713. CrossRef
Flåm, S.D. and Antipin, A.S., Equilibrium programming using proximal-like algorithms. Math. Prog. 78 (1997) 29–41. CrossRef
Flores-Bazán, F., Existence theorems for generalized noncoercive equilibrium problems: quasiconvex case. SIAM J. Optim. 11 (2000) 675–790. CrossRef
Harker, P.T. and Pang, J.S., Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications. Math. Prog. 48 (1990) 161–220. CrossRef
Hestenes, M.R., Multiplier and gradient methods. J. Optim. Theory Appl. 4 (1969) 303–320. CrossRef
J.-B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms. Springer, Berlin (1993).
Iusem, A.N., Augmented Lagrangian methods and proximal point methods for convex optimization. Investigación Operativa 8 (1999) 11–49.
Iusem, A.N. and Gárciga Otero, R., Inexact versions of proximal point and augmented Lagrangian algorithms in Banach spaces. Numer. Funct. Anal. Optim. 22 (2001) 609–640. CrossRef
Iusem, A.N., Kassay, G. and Sosa, W., On certain conditions for the existence of solutions of equilibrium problems. Math. Prog. 116 (2009) 259–273. CrossRef
Iusem, A.N. and Nasri, M., Inexact proximal point methods for equilibrium problems in Banach spaces. Numer. Funct. Anal. Optim. 28 (2007) 1279–1308. CrossRef
Iusem, A.N and Sosa, W., New existence results for equilibrium problems. Nonlinear Anal. 52 (2003) 621–635. CrossRef
Iusem, A.N. and Sosa, W., Iterative algorithms for equilibrium problems. Optimization 52 (2003) 301–316. CrossRef
A.N. Iusem and W. Sosa, On the proximal point method for equilibrium problems in Hilbert spaces in appear Optimization.
Konnov, I.V., Application of the proximal point method to nonmonotone equilibrium problems. J. Optim. Theory Appl. 119 (2003) 317–333. CrossRef
Kort, B.W. and Bertsekas, D.P., Combined primal-dual and penalty methods for convex programming. SIAM J. Control Optim. 14 (1976) 268–294. CrossRef
Krasnoselskii, M.A., Two observations about the method of succesive approximations. Uspekhi Matematicheskikh Nauk 10 (1955) 123–127.
Mastroeni, G., Gap functions for equilibrium problems. J. Glob. Optim. 27 (2003) 411–426. CrossRef
J. Moreau, Proximité et dualité dans un espace hilbertien. Bulletin de la Societé Mathématique de France 93 (1965).
Moudafi, A., Proximal point methods extended to equilibrium problems. Journal of Natural Geometry 15 (1999) 91–100.
A. Moudafi, Second-order differential proximal methods for equilibrium problems. Journal of Inequalities in Pure and Applied Mathematics 4 (2003) Article no. 18.
A. Moudafi and M. Théra, Proximal and dynamical approaches to equilibrium problems, in Ill-posed Variational Problems and Regularization Techniques, Lect. Notes in Economics and Mathematical Systems 477, Springer, Berlin (1999) 187–201.
Muu, L.D. and Oettli, W., Convergence of an adaptive penalty scheme for finding constraint equilibria. Nonlinear Anal. 18 (1992) 1159–1166. CrossRef
M. Nasri and W. Sosa, Generalized Nash games and equilibrium problems (submitted).
Noor, M.A., Auxiliary principle technique for equilibrium problems. J. Optim. Theory Appl. 122 (2004) 371–386. CrossRef
Noor, M.A. and Rassias, T.M., On nonconvex equilibrium problems. J. Math. Analysis Appl. 212 (2005) 289–299. CrossRef
M.J.D. Powell, Method for nonlinear constraints in minimization problems, in Optimization, edited by R. Fletcher, Academic Press, London (1969).
Rockafellar, R.T., A dual approach to solving nonlinear programming problems by unconstrained optimization. Math. Prog. 5 (1973) 354–373. CrossRef
Rockafellar, R.T., The multiplier method of Hestenes and Powell applied to convex programming. J. Optim. Theory Appl. 12 (1973) 555–562. CrossRef
Rockafellar, R.T., Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1 (1976) 97–116. CrossRef
Rockafellar, R.T., Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14 (1976) 877–898. CrossRef
Solodov, M.V. and Svaiter, B.F., A hybrid projection-proximal point algorithm. Journal of Convex Analysis 6 (1999) 59–70.
Solodov, M.V. and Svaiter, B.F., An inexact hybrid extragardient-proximal point algorithm using the enlargement of a maximal monotone operator. Set-Valued Analysis 7 (1999) 323–345. CrossRef