Progressive Regularization of Variational Inequalities and Decomposition Algorithms

N. El Farouq and G. Cohen
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N. El Farouq: Université de Clermont II
G. Cohen: Centre Automatique et Systèmes, École des Mines de Paris, and Scientific Advisor, INRIA

Journal of Optimization Theory and Applications, 1998, vol. 97, issue 2, No 8, 407-433

Abstract: Abstract For nonsymmetric operators involved in variational inequalities, the strong monotonicity of their possibly multivalued inverse operators (referred to as the Dunn property) appears to be the weakest requirement to ensure convergence of most iterative algorithms of resolution proposed in the literature. This implies the Lipschitz property, and both properties are equivalent for symmetric operators. For Lipschitz operators, the Dunn property is weaker than strong monotonicity, but is stronger than simple monotonicity. Moreover, it is always enforced by the Moreau–Yosida regularization and it is satisfied by the resolvents of monotone operators. Therefore, algorithms should always be applied to this regularized version or they should use resolvents: in a sense, this is what is achieved in proximal and splitting methods among others. However, the operation of regularization itself or the computation of resolvents may be as complex as solving the original variational inequality. In this paper, the concept of progressive regularization is introduced and a convergent algorithm is proposed for solving variational inequalities involving nonsymmetric monotone operators. Essentially, the idea is to use the auxiliary problem principle to perform the regularization operation and, at the same time, to solve the variational inequality in its approximately regularized version; thus, two iteration processes are performed simultaneously, instead of being nested in each other, yielding a global explicit iterative scheme. Parallel and sequential versions of the algorithm are presented. A simple numerical example demonstrates the behavior of these two versions for the case where previously proposed algorithms fail to converge unless regularization or computation of a resolvent is performed at each iteration. Since the auxiliary problem principle is a general framework to obtain decomposition methods, the results presented here extend the class of problems for which decomposition methods can be used.

Keywords: Variational equalities; optimization problems; monotonicity; regularization; Dunn property; firm contraction; convergence of algorithms (search for similar items in EconPapers)
Date: 1998
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Citations: View citations in EconPapers (3)

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DOI: 10.1023/A:1022634902457

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