Journal of Optimization Theory and Applications

, Volume 59, Issue 3, pp 369–390 | Cite as

Nested monotony for variational inequalities over product of spaces and convergence of iterative algorithms

  • G. Cohen
  • F. Chaplais
Contributed Papers


The auxiliary problem principle has been proposed by the first author as a framework to describe and analyze iterative algorithms such as gradient as well as decomposition/coordination algorithms for optimization problems (Refs. 1–3) and variational inequalities (Ref. 4). The key assumption to prove the global and strong convergence of such algorithms, as well as of most of the other algorithms proposed in the literature, is the strong monotony of the operator involved in the variational inequalities. In this paper, we consider variational inequalities defined over a product of spaces and we introduce a new property of strong nested monotony, which is weaker than the ordinary overall strong monotony generally assumed. In some sense, this new concept seems to be a minimal requirement to insure convergence of the algorithms alluded to above. A convergence theorem based on this weaker assumption is given. Application of this result to the computation of Nash equilibria can be found in another paper (Ref. 5).

Key Words

Monotony convergence of algorithms variational inequalities optimization problems decomposition coordination algorithms 


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  1. 1.
    Cohen, G.,Optimization by Decomposition and Coordination: A Unified Approach, IEEE Transactions on Automatic Control, Vol. AC-23, No. 2, pp. 222–232, 1978.Google Scholar
  2. 2.
    Cohen, G.,Auxiliary Problem Principle and Decomposition of Optimization Problems, Journal of Optimization Theory and Applications, Vol. 32, No. 3, pp. 277–305, 1980.Google Scholar
  3. 3.
    Cohen, G., andZhu, D. L.,Decomposition Coordination Methods in Large-Scale Optimization Problems: The Nondifferentiable Case and the Use of Augmented Lagrangians, Advances in Large-Scale Systems, Theory and Applications, Edited by J. B. Cruz, JAI Press, Greenwich, Connecticut, Vol. 1, pp. 203–266, 1984.Google Scholar
  4. 4.
    Cohen, G.,Auxiliary Problem Principle Extended to Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 59, No. 2, pp. 325–333, 1988.Google Scholar
  5. 5.
    Cohen, G.,Nash Equilibria: Gradient and Decomposition Algorithms, Large Scale Systems, Vol. 12, pp. 173–184, 1987.Google Scholar
  6. 6.
    Ekeland, I., andTemam R.,Convex Analysis and Variational Problems, North-Holland, Amsterdam, The Netherlands, 1976.Google Scholar
  7. 7.
    Aubin, J. P., andEkeland, I.,Applied Nonlinear Analysis, John Wiley and Sons, New York, New York, 1984.Google Scholar
  8. 8.
    Pang, J. S.,Asymmetrical Variational Inequality Problems Over Product Sets: Applications and Iterative Methods, Mathematical Programming, Vol. 31, No. 2, pp. 206–219, 1985.Google Scholar
  9. 9.
    Pang, J. S., andChan, D.,Iterative Methods for Variational and Complementarity Problems, Mathematical Programming, Vol. 24, No. 3, pp. 284–313, 1982.Google Scholar
  10. 10.
    Glowinski, R., Lions, J. L., andTremolieres, R.,Analyse Numérique des Inéquations Variationnelles, Tome 1: Théorie Générale, Premières Applications, Dunod, Paris, France, 1976.Google Scholar

Copyright information

© Plenum Publishing Corporation 1988

Authors and Affiliations

  • G. Cohen
    • 1
    • 2
  • F. Chaplais
    • 1
  1. 1.Centre d'Automatique et InformatiqueEcole Nationale Supérieure des Mines de ParisFontainebleauFrance
  2. 2.Institut National de Recherche en Informatique et AutomatiqueLe ChesnayFrance

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