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Augmented Lagrangian Theory, Duality and Decomposition Methods for Variational Inequality Problems

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Abstract

In this paper, we develop the augmented Lagrangian theory and duality theory for variational inequality problems. We propose also decomposition methods based on the augmented Lagrangian for solving complex variational inequality problems with coupling constraints.

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References

  1. MOSCO, U., Dual Variational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 40, pp. 202-206, 1972.

    Google Scholar 

  2. FUKUSHIMA, M., and ITOH, T., A Dual Approach to Asymmetric Traffic Equilibrium Problems, Mathematica Japonica, Vol. 32, pp. 701-702, 1987.

    Google Scholar 

  3. CASTELLANI, M., and MASTROENI, G., On the Duality Theory for Finite-Dimensional Variational Inequalities, Variational Inequalities and Network Equilibrium Problems, Edited by F. Giannessi and A. Maugeri, Plenum Publishing Corporation, New York, NY, pp. 21-23, 1995.

    Google Scholar 

  4. ROBINSON, S.M., Composition Duality and Maximal Monotonicity, Mathematical Programming, Vol. 85, pp. 1-13, 1999.

    Google Scholar 

  5. MARCOTTE, P., and ZHU, D. L., The Dual Theory and Decomposition Methods for Monotone Variational Inequality Problems, Preprint, 1999.

  6. ROCKAFELLAR, R.T., A Dual Approach to Solving Nonlinear Programming Problems by Unconstrained Optimization, Mathematical Programming, Vol. 5, pp. 354-373, 1973.

    Google Scholar 

  7. BERTSEKAS, D. P., Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, NY, 1982.

    Google Scholar 

  8. COHEN, G., and ZHU, 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 

  9. ZHU, D. L., and MARCOTTE, P., Cocoercivity and Its Role in the Convergence of Iterative Schemes for Solving Variational Inequalities, SIAM Journal on Optimization, Vol. 6, pp. 714-726, 1996.

    Google Scholar 

  10. ZHU, D. L., and MARCOTTE, P., New Classes of Generalized Monotonicity, Journal of Optimization Theory and Applications, Vol. 87, pp. 457-471, 1995.

    Google Scholar 

  11. BRUCK, R. E., Jr., An Iterative Solution of a Variational Inequality for Certain Monotone Operators in Hilbert Space, Bulletin of the American Mathematical Society, Vol. 81, pp. 890-892, 1975.

    Google Scholar 

  12. TSENG, P., Further Applications of a Matrix-Splitting Algorithm to Decomposition in Variational Inequalities and Convex Programming, Mathematical Programming, Vol. 48, pp. 249-264, 1990.

    Google Scholar 

  13. AUSLENDER, A., Optimization: Méthodes Numériques, Masson, Paris, France, 1973.

    Google Scholar 

  14. EKELAND, I., and TEMAN, R., Convex Analysis and Variational Problems, North Holland, Amsterdam, Netherlands, 1976.

    Google Scholar 

  15. GABAY, D., Applications of Method of Multipliers to Variational Inequalities, Augmented Lagrangian Methods: Application to the Numerical Solution of Boundary-Value Problems, Edited by M. Fortin and R. Glowinski, North Holland, Amsterdam, Netherlands, 1983.

    Google Scholar 

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Zhu, D. Augmented Lagrangian Theory, Duality and Decomposition Methods for Variational Inequality Problems. Journal of Optimization Theory and Applications 117, 195–216 (2003). https://doi.org/10.1023/A:1023664709604

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