Augmented Lagrangian methods for variational inequality problems

Alfredo N. Iusem; Mostafa Nasri

RAIRO - Operations Research (2010)

  • Volume: 44, Issue: 1, page 5-25
  • ISSN: 0399-0559

Abstract

top
We introduce augmented Lagrangian methods for solving finite dimensional variational inequality problems whose feasible sets are defined by convex inequalities, generalizing the proximal augmented Lagrangian method for constrained optimization. At each iteration, primal variables are updated by solving an 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.

How to cite

top

Iusem, Alfredo N., and Nasri, Mostafa. "Augmented Lagrangian methods for variational inequality problems." RAIRO - Operations Research 44.1 (2010): 5-25. <http://eudml.org/doc/250859>.

@article{Iusem2010,
abstract = { We introduce augmented Lagrangian methods for solving finite dimensional variational inequality problems whose feasible sets are defined by convex inequalities, generalizing the proximal augmented Lagrangian method for constrained optimization. At each iteration, primal variables are updated by solving an 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. },
author = {Iusem, Alfredo N., Nasri, Mostafa},
journal = {RAIRO - Operations Research},
keywords = {Augmented Lagrangian method; equilibrium problem; inexact solution; proximal point method; variational inequality problem},
language = {eng},
month = {2},
number = {1},
pages = {5-25},
publisher = {EDP Sciences},
title = {Augmented Lagrangian methods for variational inequality problems},
url = {http://eudml.org/doc/250859},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Iusem, Alfredo N.
AU - Nasri, Mostafa
TI - Augmented Lagrangian methods for variational inequality problems
JO - RAIRO - Operations Research
DA - 2010/2//
PB - EDP Sciences
VL - 44
IS - 1
SP - 5
EP - 25
AB - We introduce augmented Lagrangian methods for solving finite dimensional variational inequality problems whose feasible sets are defined by convex inequalities, generalizing the proximal augmented Lagrangian method for constrained optimization. At each iteration, primal variables are updated by solving an 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.
LA - eng
KW - Augmented Lagrangian method; equilibrium problem; inexact solution; proximal point method; variational inequality problem
UR - http://eudml.org/doc/250859
ER -

References

top
  1. A.S. Antipin, Equilibrium programming: proximal methods. Comput. Math. Math. Phys.37 (1997) 1285–1296.  
  2. A.S. Antipin, F.P. Vasilev and A.S. Stukalov, A regularized Newton method for solving equilibrium programming problems with an inexactly specified set. Comput. Math. Math. Phys.47 (2007) 19–31.  
  3. A. Auslender and M. Teboulle, Lagrangian duality and related multiplier methods for variational inequality problems. SIAM J. Optim.10 (2000) 1097–1115.  
  4. D.P. Bertsekas, On penalty and multiplier methods for constrained optimization problems. SIAM J. Control Optim.14 (1976) 216–235.  
  5. M. Bianchi and R. Pini, Coercivity conditions for equilibrium problems. J. Optim. Theory Appl.124 (2005) 79–92.  
  6. M. Bianchi and S. Schaible, Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl.90 (1996) 31–43.  
  7. E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems. The Mathematics Student63 (1994) 123–145.  
  8. H. Brezis, L. Nirenberg and S. Stampacchia, A remark on Ky Fan minimax principle. Bolletino della Unione Matematica Italiana6 (1972) 293–300.  
  9. J.D. Buys, Dual algorithms for constrained optimization problems, Ph.D. thesis, University of Leiden, The Netherlands (1972).  
  10. F. Facchinei and J.S. Pang, Finite-dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin (2003).  
  11. M.C. Ferris and J.S. Pang, Engineering and economic applications of complementarity problems. SIAM Rev.39 (1997) 669–713.  
  12. S.D. Flåm and A.S. Antipin, Equilibrium programming using proximal-like algorithms. Math. Prog.78 (1997) 29–41.  
  13. F. Flores-Bazán, Existence theorems for generalized noncoercive equilibrium problems: quasiconvex case. SIAM J. Optim.11 (2000) 675–790.  
  14. P.T. Harker and J.S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications. Math. Prog.48 (1990) 161–220.  
  15. M.R. Hestenes, Multiplier and gradient methods. J. Optim. Theory Appl.4 (1969) 303–320.  
  16. J.-B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms. Springer, Berlin (1993).  
  17. A.N. Iusem, Augmented Lagrangian methods and proximal point methods for convex optimization. Investigación Operativa8 (1999) 11–49.  
  18. A.N. Iusem and R. Gárciga Otero, Inexact versions of proximal point and augmented Lagrangian algorithms in Banach spaces. Numer. Funct. Anal. Optim.22 (2001) 609–640.  
  19. A.N. Iusem, G. Kassay and W. Sosa, On certain conditions for the existence of solutions of equilibrium problems. Math. Prog.116 (2009) 259–273.  
  20. A.N. Iusem and M. Nasri, Inexact proximal point methods for equilibrium problems in Banach spaces. Numer. Funct. Anal. Optim.28 (2007) 1279–1308.  
  21. A.N Iusem and W. Sosa, New existence results for equilibrium problems. Nonlinear Anal.52 (2003) 621–635.  
  22. A.N. Iusem and W. Sosa, Iterative algorithms for equilibrium problems. Optimization52 (2003) 301–316.  
  23. A.N. Iusem and W. Sosa, On the proximal point method for equilibrium problems in Hilbert spaces in appear Optimization.  
  24. I.V. Konnov, Application of the proximal point method to nonmonotone equilibrium problems. J. Optim. Theory Appl.119 (2003) 317–333.  
  25. B.W. Kort and D.P. Bertsekas, Combined primal-dual and penalty methods for convex programming. SIAM J. Control Optim.14 (1976) 268–294.  
  26. M.A. Krasnoselskii, Two observations about the method of succesive approximations. Uspekhi Matematicheskikh Nauk10 (1955) 123–127.  
  27. G. Mastroeni, Gap functions for equilibrium problems. J. Glob. Optim.27 (2003) 411–426.  
  28. J. Moreau, Proximité et dualité dans un espace hilbertien. Bulletin de la Societé Mathématique de France93 (1965).  
  29. A. Moudafi, Proximal point methods extended to equilibrium problems. Journal of Natural Geometry15 (1999) 91–100.  
  30. A. Moudafi, Second-order differential proximal methods for equilibrium problems. Journal of Inequalities in Pure and Applied Mathematics4 (2003) Article no. 18.  
  31. 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 Systems477, Springer, Berlin (1999) 187–201.  
  32. L.D. Muu and W. Oettli, Convergence of an adaptive penalty scheme for finding constraint equilibria. Nonlinear Anal.18 (1992) 1159–1166.  
  33. M. Nasri and W. Sosa, Generalized Nash games and equilibrium problems (submitted).  
  34. M.A. Noor, Auxiliary principle technique for equilibrium problems. J. Optim. Theory Appl.122 (2004) 371–386.  
  35. M.A. Noor and T.M. Rassias, On nonconvex equilibrium problems. J. Math. Analysis Appl.212 (2005) 289–299.  
  36. M.J.D. Powell, Method for nonlinear constraints in minimization problems, in Optimization, edited by R. Fletcher, Academic Press, London (1969).  
  37. R.T. Rockafellar, A dual approach to solving nonlinear programming problems by unconstrained optimization. Math. Prog.5 (1973) 354–373.  
  38. R.T. Rockafellar, The multiplier method of Hestenes and Powell applied to convex programming. J. Optim. Theory Appl.12 (1973) 555–562.  
  39. R.T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res.1 (1976) 97–116.  
  40. R.T. Rockafellar, Monotone operators and the proximal point algorithm. SIAM J. Control Optim.14 (1976) 877–898.  
  41. M.V. Solodov and B.F. Svaiter, A hybrid projection-proximal point algorithm. Journal of Convex Analysis6 (1999) 59–70.  
  42. M.V. Solodov and B.F. Svaiter, An inexact hybrid extragardient-proximal point algorithm using the enlargement of a maximal monotone operator. Set-Valued Analysis7 (1999) 323–345.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.