Abstract
We consider general variational inequalities with a multivalued maximal monotone operator in a Hilbert space. For solving these problems, Cohen developed several years ago the auxiliary problem method. Perturbed versions of this method have been already studied in the literature for the single-valued case. They allow to consider for example, barrier functions and interior approximations of the feasible domain. In this paper, we present a relaxation of these perturbation methods by using the concept of ε-enlargement of a maximal monotone operator. We prove that, under classical assumptions, the sequence generated by this scheme is bounded and weakly convergent to a solution of the problem. Strong convergence is also obtained under additional conditions.
In the particular case of nondifFerentiable convex optimization, the ε-subdifferential will take place of the ε-enlargement and some assumptions for convergence will be weakened. In the nonperturbed situation, our scheme reduces to the projected inexact subgradient procedure.
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Salmon, G., Nguyen, V.H., Strodiot, JJ. (2000). A Perturbed and Inexact Version of the Auxiliary Problem Method for Solving General Variational Inequalities with a Multivalued Operator. In: Nguyen, V.H., Strodiot, JJ., Tossings, P. (eds) Optimization. Lecture Notes in Economics and Mathematical Systems, vol 481. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57014-8_27
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DOI: https://doi.org/10.1007/978-3-642-57014-8_27
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