Abstract
In recent years, the so-called auxiliary problem principle has been used to derive many iterative type algorithms for solving optimal control, mathematical programming, and variational inequality problems. In the present paper, we use this principle in conjunction with the epiconvergence theory to introduce and study a general family of perturbation methods for solving nonlinear variational inequalities over a product space of reflexive Banach spaces. We do not assume that the monotone operator involved in our general variational inequality problem is of potential type. Several known iterative algorithms, which can be obtained from our theory, are also discussed.
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Ekeland, I., andTemam, R.,Convex Analysis and Variational Inequalities, North Holland, Amsterdam, Holland, 1976.
Cohen, G., andChaplais, 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.
Auslender, A.,Optimisation: Méthodes Numériques, Masson, Paris, France, 1976.
Glowinski, R., Lions, J. L., andTrémolières, R.,Numerical Analysis of Variational Inequalities, North Holland, Amsterdam, Holland, 1981.
Stampacchia, G.,Regularity of Solutions of Some Variational Inequalities, Proceedings in Pure Mathematics, Vol. 28, pp. 271–281, 1970.
Dafermos, S.,An Iterative Scheme for Variational Inequalities, Mathematical Programming, Vol. 26, pp. 40–47, 1983.
Pang, J. S.,Asymmetric Variational Inequality Problems over Product Sets: Applications and Iterative Methods, Mathematical Programming, Vol. 31, pp. 206–219, 1985.
Marcotte, P., andDussault, J. P.,A Sequential Linear Programming Algorithm for Solving Monotone Variational Inequalities, SIAM Journal on Control and Optimization, Vol. 27, pp. 1260–1278, 1989.
Tseng, P.,Further Applications of a Splitting Algorthm to Decomposition in Variational Inequalities and Convex Programming, Mathematical Programming, vol. 48, pp. 249–263, 1990.
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.
Taji, K., Fukushima, M., andIbaraki, T.,A Globally Convergent Newton Method for Solving Strongly Monotone Variational Inequalities, Mathematical Programming, Vol. 58, pp. 369–383, 1993.
Karamardian, S.,Generalized Complementarity Problem, Journal of Optimization Theory and Applications, Vol. 8, pp. 161–168, 1971.
Isac, G.,Complementarity Problems, Lecture Notes in Mathematics, Springer Verlag, Heidelberg, Germany, Vol. 1528, 1992.
Cohen, G.,Auxiliary Problem Principle and Decomposition of Optimization Problems, Journal of Optimization Theory and Applications, Vol. 32, pp. 277–305, 1980.
Han, S. P., andLou, G.,A Parallel Algorithm for a Class of Convex Programs, SIAM Journal on Control and Optimization, Vol. 26, pp. 345–355, 1988.
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.
Mouallif, K., Nguyen, V. H., andStrodiot, 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.
Harker, P. T., andPang, 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.
Attouch, H.,Variational Convergence for Functions and Operators, Pitman, London, England, 1984.
Cohen, G.,Auxiliary Problem Principle Extended to Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 59, pp. 325–333, 1988.
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, Edited by J. B. Cruz Jr., JAI Press, Greenwich, Connecticut, Vol. 1, pp. 203–266, 1984.
Harker, P. T., andXiao, B.,Newton's Method for the Nonlinear Complementarity Problem: A B-Differentiable Equation Approach, Mathematical Programming, Vol. 48, pp. 339–357, 1990.
Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1972.
Lions, J. L.,Quelques Méthodes de Résolution de Problèmes aux Limites Non Linéaires Dunod, Paris, France, 1969.
Attouch, H., andWets, 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.
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.
Guillet, A.,Méthodes de Pénalisation pour la Résolution d'Inégalités Variationnelles, Technical Report, Department of Applied Mathematics, Université Blaise Pascal, Clermont-Ferrand, France, 1974.
Isac, G.,Nonlinear Complementarity Problem and Galerkin Method, Journal of Mathematical Analysis and Applications, Vol. 108, pp. 563–574, 1985.
Pang, J. S., andChan, D.,Iterative Methods for Variational and Complementarity Problems, Mathematical Programming, Vol. 24, pp. 284–317, 1982.
Marcotte, P., andDussault, J. P.,A Note on a Globally Convergent Newton Method for Solving Monotone Variational Inequalities, Operations Research Letters, Vol. 6, pp. 35–43, 1987.
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.
Auslender, A., Crouzeix, J. P., andFedit, P.,Penalty-Proximal Methods in Convex Programming, Journal of Optimization Theory and Applications, Vol. 55, pp. 1–21, 1987.
Rockafellar, R. T.,Monotone Operators and the Proximal Point Algorithm, SIAM Journal on Control and Optimization, Vol. 14, pp. 877–898, 1976.
Pascali, D., andSburlan, S.,Nonlinear Mappings of Monotone Type, Editura Academiei, Bucarest, Romania, 1978.
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.
Ortega, J. M., andRheinboldt, W. C.,Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970.
Makler-Scheimberg, S., Nguyen, V. H., andStrodiot, 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.
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Communicated by J. P. Crouzeix
This work was completed while the second author was visiting the Department of Mathematics of the University of Washington, Seattle, Washington under financial support from the Belgian Fonds National de la Recherche Scientifique, Grant FNRS: B8/5-JS-9. 549.
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Makler-Scheimberg, S., Nguyen, V.H. & Strodiot, J.J. Family of perturbation methods for variational inequalities. J Optim Theory Appl 89, 423–452 (1996). https://doi.org/10.1007/BF02192537
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DOI: https://doi.org/10.1007/BF02192537