An interior point method with Bregman functions for the variational inequality problem with paramonotone operators

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• 作者：Censor ; Yair ; Iusem ; Alfredo N. ; Zenios ; Stavros A.
• 关键词：Variational inequalities ; Monotone operators ; Paramonotone operators ; Convex programming ; Generalized distances
• 刊名：Mathematical Programming
• 出版年：1998
• 出版时间：May 1, 1998
• 年：1998
• 卷：81
• 期：3
• 页码：373-400
• 全文大小：1.21 M

文摘

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.