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Zbl 0919.90123
Censor, Yair; Iusem, Alfredo N.; Zenios, Stavros A.
An interior point method with Bregman functions for the variational inequality problem with paramonotone operators. (English)
[J] Math. Program. 81, No.3 (A), 373-400 (1998). ISSN 0025-5610; ISSN 1436-4646/e

Summary: We present an algorithm for the variational inequality problem on convex sets with nonempty interior. The use of Bregman functions whose zone is the convex set allows for the generation of a sequence contained in the interior, without taking explicitly into account the constraints which define the convex set. We establish full convergence to a solution with minimal conditions upon the monotone operator $F$, weaker than strong monotonicity or Lipschitz continuity, for instance, and including cases where the solution needs not be unique. We apply our algorithm to several relevant classes of convex sets, including orthants, boxes, polyhedra and balls, for which Bregman functions are presented which give rise to explicit iteration formulae, up to the determination of two scalar stepsizes, which can be found through finite search procedures.

MSC 2000:: *90C25 Convex programming

Keywords: paramonotone operators; interior point method; variational inequality; Bregman functions; monotone operator; generalized distances

Cited in: Zbl 1153.90566 Zbl 1133.49010

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