Family of Perturbation Methods for Variational Inequalities

Journal of Optimization Theory and Applications

Volumn 89, Issue 2, 1996, Pages 423-452

  Makler Scheimberg, S ^a   Nguyen, V H ^b   Strodiot, J J ^b  

^a FEDERAL UNIVERSITY OF RIO DE JANEIRO   (Brazil)

^b UNIVERSITY OF NAMUR   (Belgium)

Author keywords

Auxiliary problem principle; Decomposition algorithms; Epiconvergence; Monotone operators; Nonlinear complementarity; Variational inequalities

Indexed keywords

BANACH SPACES; ITERATIVE METHODS; PERTURBATION TECHNIQUES; VARIATIONAL TECHNIQUES;

ALGORITHM FOR SOLVING; AUXILIARY PROBLEM PRINCIPLE; DECOMPOSITION ALGORITHM; EPI-CONVERGENCE; MONOTONE OPERATORS; NONLINEAR COMPLEMENTARITY; OPTIMAL CONTROLS; PERTURBATION METHOD; VARIATIONAL INEQUALITIES; VARIATIONAL INEQUALITY PROBLEMS;

MATHEMATICAL PROGRAMMING;

EID: 0002250027     PISSN: 00223239     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02192537     Document Type: Article

References (37)

1. EKELAND, I., and TEMAM, R., Convex Analysis and Variational Inequalities, North Holland, Amsterdam, Holland, 1976.
2. COHEN, G., and CHAPLAIS, F., Nested Monotonicity for Variational Inequalities over Product of Spaces and Convergence of Iterative Algorithms, Journal of Optimization Theory and Applications, Vol. 59, pp. 369-390, 1988.
3. AUSLENDER, A., Optimisation: Méthodes Numériques, Masson, Paris, France, 1976.
4. GLOWINSKI, R., LIONS, J. L., and TRÉMOLIÊRES, R., Numerical Analysis of Variational Inequalities, North Holland, Amsterdam, Holland, 1981.
5. STAMPACCHIA, G., Regularity of Solutions of Some Variational Inequalities, Proceedings in Pure Mathematics, Vol. 28, pp. 271-281, 1970.
6. DAFERMOS, S., An Iterative Scheme for Variational Inequalities, Mathematical Programming, Vol. 26, pp. 40-47, 1983.
7. PANG, J. S., Asymmetric Variational Inequality Problems over Product Sets: Applications and Iterative Methods, Mathematical Programming, Vol. 31, pp. 206-219, 1985.
8. MARCOTTE, P., and DUSSAULT, J. P., A Sequential Linear Programming Algorithm for Solving Monotone Variational Inequalities, SIAM Journal on Control and Optimization, Vol. 27, pp. 1260-1278, 1989.
9. TSENG, P., Further Applications of a Splitting Algorithm to Decomposition in Variational Inequalities and Convex Programming, Mathematical Programming, Vol. 48, pp. 249-263, 1990.
10. TSENG, P., Applications of a Splitting Algorithm to Decomposition in Convex Programming and Variational Inequalities, SIAM Journal on Control and Optimization, Vol. 29, pp. 119-138, 1991.
11. TAJI, K., FUKUSHIMA, M., and IBARAKI, T., A Globally Convergent Newton Method for Solving Strongly Monotone Variational Inequalities, Mathematical Programming, Vol. 58, pp. 369-383, 1993.
12. KARAMARDIAN, S., Generalized Complementarity Problem, Journal of Optimization Theory and Applications, Vol. 8, pp. 161-168, 1971.
13. ISAC, G., Complementarity Problems, Lecture Notes in Mathematics, Springer Verlag, Heidelberg, Germany, Vol. 1528, 1992.
14. COHEN, G., Auxiliary Problem Principle and Decomposition of Optimization Problems, Journal of Optimization Theory and Applications, Vol. 32, pp. 277-305, 1980.
15. HAN, S. P., and LOU, G., A Parallel Algorithm for a Class of Convex Programs, SIAM Journal on Control and Optimization, Vol. 26, pp. 345-355, 1988.
16. LEMAIRE, B., Coupling Optimization Methods and Variational Convergence, Trends in Mathematical Optimization, Edited by K. H. Hoffman, J. B. Hiriart-Urruty, C. Lemaréchal, and J. Zowe, Birkhäuser Verlag, Basel, Switzerland, pp. 163-179, 1988.
17. MOUALLIF, K., NGUYEN, V. H., and STRODIOT, J. J., A Perturbed Parallel Decomposition Method for a Class of Nonsmooth Convex Minimization Problems, SIAM Journal on Control and Optimization, Vol. 29, pp. 829-847, 1991.
18. HARKER, P. T., and PANG, J. S., Finite-Dimensional Variational Inequality and Nonlinear Complementarity Problems: A Survey of Theory, Algorithms and Applications, Mathematical Programming, Vol. 48, pp. 161-220, 1990.
19. ATTOUCH, H., Variational Convergence for Functions and Operators, Pitman, London, England, 1984.
20. COHEN, G., Auxiliary Problem Principle Extended to Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 59, pp. 325-333, 1988.
21. 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, Edited by J. B. Cruz, Jr., JAI Press, Greenwich, Connecticut, Vol. 1, pp. 203-266, 1984.
22. HARKER, P. T., and XIAO, B., Newton's Method for the Nonlinear Complementarity Problem: A B-Differentiable Equation Approach, Mathematical Programming, Vol. 48, pp. 339-357, 1990.
23. ROCKAFELLAR, R. T., Convex Analysis, Princeton University Press, Princeton, New Jersey, 1972.
24. LIONS, J. L., Quelques Méthodes de Résolution de Problèmes aux Limites Non Linéaires, Dunod, Paris, France, 1969.
25. ATTOUCH, H., and WETS, R. J. B., Approximation and Convergence in Nonlinear Optimization, Nonlinear Programming 4, Edited by O. L. Mangasarian, R. R. Meyer, and S. M. Robinson, Academic Press, New York, New York, pp. 367-394, 1981.
26. GWINNER, J., On the Penalty for Constrained Variational Inequalities, Optimization: Theory, Algorithms and Applications, Edited by J. B. Hiriart-Urruty, W. Oettli, and J. Stoer, Marcel Dekker, New York, New York, pp. 197-211, 1983.
27. GUILLET, A., Méthodes de Pénalisation pour la Résolution d'Inégalités Variationnelles, Technical Report, Department of Applied Mathematics, Université Blaise Pascal, Clermont-Ferrand, France, 1974.
28. ISAC, G., Nonlinear Complementarity Problem and Galerkin Method, Journal of Mathematical Analysis and Applications, Vol. 108, pp. 563-574, 1985.
29. PANG, J. S., and CHAN, D., Iterative Methods for Variational and Complementarity Problems, Mathematical Programming, Vol. 24, pp. 284-317, 1982.
30. MARCOTTE, P., and DUSSAULT, J. P., A Note on a Globally Convergent Newton Method for Solving Monotone Variational Inequalities, Operations Research Letters, Vol. 6, pp. 35-43, 1987.
31. AUSLENDER, A., Numerical Methods for Nondifferentiable Convex Optimization, Mathematical Programming Study, Edited by B. Cornet, V. H. Nguyen, and J. P. Vial, North Holland, Amsterdam, Holland, Vol. 30, pp. 102-126, 1987.
32. AUSLENDER, A., CROUZEIX, J. P., and FEDIT, P., Penalty-Proximal Methods in Convex Programming, Journal of Optimization Theory and Applications, Vol. 55, pp. 1-21, 1987.
33. ROCKAFELLAR, R. T., Monotone Operators and the Proximal Point Algorithm, SIAM Journal on Control and Optimization, Vol. 14, pp. 877-898, 1976.
34. PASCALI, D., and SBURLAN, S., Nonlinear Mappings of Monotone Type, Editura Academiei, Bucarest, Romania, 1978.
35. WETS, R. J. B., Convergence of Convex Functions, Variational Inequalities and Convex Optimization Problems, Variational Inequalities and Complementarity Problems, Edited by R. W. Cottle, F. Giannessi, and J. L. Lions, Wiley, Chichester, England, pp. 376-403, 1980.
36. ORTEGA, J. M., and RHEINBOLDT, W. C., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970.
37. MAKLER-SCHEIMBERG, S., NGUYEN, V. H., and STRODIOT, J. J., A Family of Perturbed Parallel Decomposition Methods for Variational Inequalities, Report 94-10, Department of Mathematics, Facultés Universitaires Notre Dame de la Paix, Namur, Belgium, 1994.