On Some Properties of Generalized Proximal Point Methods for Variational Inequalities

A. N. Iusem
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A. N. Iusem: Researcher, Institute de Matemática Pura e Aplicada

Journal of Optimization Theory and Applications, 1998, vol. 96, issue 2, No 5, 337-362

Abstract: Abstract We discuss here generalized proximal point methods applied to variational inequality problems. These methods differ from the classical point method in that a so-called Bregman distance substitutes for the Euclidean distance and forces the sequence generated by the algorithm to remain in the interior of the feasible region, assumed to be nonempty. We consider here the case in which this region is a polyhedron (which includes linear and nonlinear programming, monotone linear complementarity problems, and also certain nonlinear complementarity problems), and present two alternatives to deal with linear equality constraints. We prove that the sequences generated by any of these alternatives, which in general are different, converge to the same point, namely the solution of the problem which is closest, in the sense of the Bregman distance, to the initial iterate, for a certain class of operators. This class consists essentially of point-to-point and differentiable operators such that their Jacobian matrices are positive semidefinite (not necessarily symmetric) and their kernels are constant in the feasible region and invariant through symmetrization. For these operators, the solution set of the problem is also a polyhedron. Thus, we extend a previous similar result which covered only linear operators with symmetric and positive-semidefinite matrices.

Keywords: Proximal point methods; generalized distances; variational inequalities; interior point methods; linear complementarity problems; nonlinear complementarity problems (search for similar items in EconPapers)
Date: 1998
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Citations: View citations in EconPapers (2)

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DOI: 10.1023/A:1022670114963

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