Abstract
In this paper, we develop a very general descent framework for solving asymmetric, monotone variational inequalities. We introduce two classes of differentiable merit functions and the associated global convergence frameworks which include, as special instances, the projection, Newton, quasi-Newton, linear Jacobi, and nonlinear methods. The generic algorithm is very flexible and consequently well suited for exploiting any particular structure of the problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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.
Dafermos, S.,An Iterative Scheme for Variational Inequalities, Mathematical Programming, Vol. 26, pp. 40–47, 1983.
Zanni, L.,On the Convergence Rate of Two Projection Methods for Variational Inequalities in R n, Calcolo, 1992.
Cohen, G.,Auxiliary Problem Principle Extended to Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 59, pp. 325–333, 1988.
Marcotte, P.,A New Algorithm for Solving Variational Inequalities, with Application to the Traffic Assignment Problem, Mathematical Programming, Vol. 33, pp. 339–351, 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.
Marcotte, P., andDussault, J. P.,A Note on a Globally Convergent Newton Method for Solving Monotone Variational Inequalities, Operations Research Letters, Vol. 6, pp. 35–42, 1987.
Fukushima, M.,Equivalent Differentiable Optimization Problems and Descent Methods for Asymmetric Variational Inequality Problems, Mathematical Programming, Vol. 53, pp. 99–110, 1992.
Wu, J. H., Florian, M., andMarcotte, P.,A General Descent Framework for the Monotone Variational Inequality Problem, Mathematical Programming, Vol. 61, pp. 281–300 1993.
Taji, K., Fukushima, M., andIbaraki, T.,A Globally Convergent Newton Method for Solving Monotone Variational Inequalities, Mathematical Programming, Vol. 58, pp. 369–383, 1993.
Zhu, D. L., andMarcotte, P.,Auxiliary Problem Principle and Descent Methods, Preprint, University of Montreal, 1991.
Author information
Authors and Affiliations
Additional information
Communicated by F. Giannessi
This research was supported by the National Science and Engineering Research Council of Canada, Grant A5789, and by the Department of National Defence of Canada, Grant FUHBP.
Rights and permissions
About this article
Cite this article
Zhu, D.L., Marcotte, P. An extended descent framework for variational inequalities. J Optim Theory Appl 80, 349–366 (1994). https://doi.org/10.1007/BF02192941
Issue Date:
DOI: https://doi.org/10.1007/BF02192941