Abstract
The purpose of this paper is to study the continuous dependence of solutions of variational inequalities with respect to perturbations of the data that are maximal monotone operators and closed convex functions. The constraint sets are defined by a finite number of linear equalities and non linear convex inequalities. Primal and dual stability results are given, extending the classical ones for optimization problems.
Similar content being viewed by others
References
H. Attouch, Z. Chbani, and A. Moudafi, “Recession operators and solvability of variational problems,” Series on Advances in Mathematics for Applied Sciences, Vol. 18, World Scientif. 1994, pp. 51-67.
A. Auslender, “Convergence of stationary sequences for variational inequalities with maximal monotone operators,” Appl. Math. Optimization, vol. 28, pp. 161-172, 1993.
A. Auslender, “Asymptotic analysis for penalty and barrier methods in noncoercive variational inequalities,” SIAM J. Control and Optimization, accepted for publication.
A. Auslender and M. Teboulle, “Modified Lagrangian methods for variational inequalities,” SIAM Journal on Optimization, January 1999, submitted.
C. Baiocchi, G. Buttazo, F. Gastaldi, and F. Tomarelli, “General existence theorems for unilateral problems in continuum mechanics,” Archives Rational Mechanics and Analysis, pp. 149-189, 1988.
B. Bank, J. Guddat, D. Klatte, B. Kunmer, and K. Tammer, Non-Linear Parametric Optimization, Akademic-Vermag: Berlin, 1982.
A. Bensoussan, J. Lions, and R. Temam, “Sur les méthodes de décomposition, de décentralisation, de coordinations et applications,” in Methodes Numériques en Sciences Physiques et Economiques, J.L. Lions and G.I. Marchouk (Eds.), Dunod Bordas: Paris, 1974, pp. 133-257.
H. Brezis and L. Nirenberg, “Characterization of ranges of some linear operators and applications to buondary value problems,” Ann. Scuola Norm. Sup Pisa Cl. Sci., vol. 4, p.5, 1978.
F. Browder, “On the range of the sum of nonlinear operators and the Landesman-Lazer principle,” Bull. Uni. Mat. Ital. B, vol. 5, pp. 364-376, 1979.
Richard W. Cottle, Jong-Shi Pang, and Richard E. Stone, The linear complementary problem, Computer Science and Scientific Computing, Academic-Press Inc., 1992.
Anthony V. Fiacco and Jiming Liu, “On the stability of general convex programs under Slater's condition and primal solution boundedness,” Optimization, vol. 32, no. 4, pp. 291-299, 1995.
P.L. Lions, “Two remarks on the convergence of convex functions and monotone operators,” Nonlinear Analy., vol. 2, pp. 553-562, 1978.
R.T. Rockafellar, Convex Analysis, Princeton University Press: Princeton, NJ, 1970.
R.T Rockafellar, “Monotone operators and augmented Lagrangian in nonlinear programming 3,” Academic Press: New York, pp. 1-25, 1978; Siam J. of Control and Optimization, vol., 14, pp. 877-898, 1976.
R.T. Rockafellar and R.J.B Wets, Variational Analysis, Springer Verlag, 1998.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Auslender, A., Correa, R. Primal and Dual Stability Results for Variational Inequalities. Computational Optimization and Applications 17, 117–130 (2000). https://doi.org/10.1023/A:1026594114013
Issue Date:
DOI: https://doi.org/10.1023/A:1026594114013