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Journal of Convex Analysis 18 (2011), No. 3, 601--619
Copyright Heldermann Verlag 2011

On the Cutting Plane Property and the Bregman Proximal Point Algorithm

Nils Langenberg
Dept. of Mathematics, University of Trier, 54286 Trier, Germany

The Bregman-function-based Proximal Point Algorithm (BPPA) for solving variational inequalities is considered. In this framework the customary assumption of the cutting plane property (CPP) of the related operator is investigated. Since this property cannot be expected in saddle-point-problems, it should be considered as rather restrictive. This paper contributes to the situation when the CPP fails to hold. For this situation, interior proximal(-like) methods have only been constructed for polyhedral sets up to now.

Under very mild assumptions (not implying the CPP) we show that whenever the sequence of iterates (generated by the BPPA) is convergent, its limit can only be a solution of the given problem. Further, using a (known) slight modification of Bregman functions, we show -- still without using the CPP -- that the sequence generated by the BPPA is convergent to a solution when the feasible set has some special nonlinear structure like a ball.

Keywords: Pseudomonotone operators, variational inequalities, Bregman distances, Proximal Point Algorithm, cutting plane property, nonconvex optimization.

MSC: 47J20, 65J20, 65K10, 90C26, 90C30

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