Abstract
Given a point-to-set operator T, we introduce the operator Tε defined as Tε(x)= {u: 〈 u − v, x − y 〉 ≥ −ε for all y ɛ Rn, v ɛ T(y)}. When T is maximal monotone Tε inherits most properties of the ε-subdifferential, e.g. it is bounded on bounded sets, Tε(x) contains the image through T of a sufficiently small ball around x, etc. We prove these and other relevant properties of Tε, and apply it to generate an inexact proximal point method with generalized distances for variational inequalities, whose subproblems consist of solving problems of the form 0 ɛ Hε(x), while the subproblems of the exact method are of the form 0 ɛ H(x). If εk is the coefficient used in the kth iteration and the εk's are summable, then the sequence generated by the inexact algorithm is still convergent to a solution of the original problem. If the original operator is well behaved enough, then the solution set of each subproblem contains a ball around the exact solution, and so each subproblem can be finitely solved.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alber, Ya. I.: On the regularization method for variational inequalities with nonsmooth unbounded operators in a Banach space, Appl. Math. Lett. 6(4) (1993), 63–68.
Brézis, H.: Opérateurs monotones maximaux et semigroups de contractions dans les espaces de Hilbert, Math. Studies 5, North-Holland, New York, 1973.
Bregman, L. M.: The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex programming, U.S.S.R. Comput. Math. and Math. Phys. 7(3) (1967), 200–217.
Burachik, R. S. and Iusem, A. N.: A generalized proximal point algorithm for the variational inequality problem in a Hilbert space, SIAM J. Optim. (to appear).
Censor, Y., Iusem, A. N. and Zenios, S. A.: An interior point method with Bregman functions for the variational inequality problem with paramonotone operators (submitted for publication).
De Pierro, A. R. and Iusem, A. N.: A relaxed version of Bregman's method for convex programming, J. Optim. Theory Appl. 51 (1986), 421–440.
Ermol'ev, Yu. M.: On the method of generalized stochastic gradients and quasi-Fejér sequences, Cybernetics 5 (1969), 208–220.
Hiriart-Urruty, J.-B. and Lemarechal, C.: Convex Analysis and Minimization Algorithms, Springer-Verlag, Berlin, 1993.
Iusem, A. N.: An iterative algorithm for the variational inequality problem, Comput. Appl. Math. 13 (1994), 103–114.
Iusem, A. N., Svaiter, B. F. and Teboulle, M.: Entropy-like proximal methods in convex programming, Math. Oper. Res. 19 (1994), 790–814.
Iusem, A. N.: On some properties of paramonotone operators (submitted for publication).
Kabbadj, S.: Méthodes proximales entropiques, Thesis in Mathematics, Université de Montpellier, France, 1994.
Kiwiel, K. C.: Methods of Descent for Nondifferentiable Optimization, Lecture Notes in Math. 1133, Springer-Verlag, Berlin, 1985.
Kiwiel, K. C.: Proximal minimization methods with generalized Bregman functions, SIAM J. Control and Optim. (to appear).
Korpelevich, G. M.: The extragradient method for finding saddle points and other problems, Ekonom. Mat. Metody 12 (1976), 747–756.
Lemaire, B.: Bounded diagonally stationary sequences in convex optimization, J. Convex Anal. 1 (1994), 75–86.
Lemaire, B.: On the convergence of some iterative methods in convex analysis, In Lecture Notes in Econom. and Math. Systems 129, Springer-Verlag, Berlin, 1995, pp. 252–268.
Liskovets, O. A.: Regularization of problems with discontinuous monotone, arbitrarily perturbed operators, Soviet Math., Dokl. 28 (1983), 324–327.
Liskovets, O. A.: Discrete regularization of problems with arbitrarily perturbed monotone operators, Soviet Math. Dokl. 34 (1987), 198–201.
Martinet, B.: Algorithmes pour la résolution de problèmes d'optimisation et de minimax, Thésed'Etát, Université de Grenoble, France, 1972.
Rockafellar, R. T.: Local boundedness of nonlinear monotone operators, Michigan Math. J. 16 (1969), 397–407.
Rockafellar, R. T.: Monotone operators and the proximal point algorithm, SIAM J. Control and Optimization 14 (1976), 877–898.
Zeidler, E.: Functional Analysis and Its Applications, Part II/B (Nonlinear Monotone Operators), Springer-Verlag, Berlin, 1985.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Burachik, R.S., Iusem, A.N. & Svaiter, B.F. Enlargement of Monotone Operators with Applications to Variational Inequalities. Set-Valued Analysis 5, 159–180 (1997). https://doi.org/10.1023/A:1008615624787
Issue Date:
DOI: https://doi.org/10.1023/A:1008615624787