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UBC Theses and Dissertations

Subgradient projectors : theory, extensions and algorithms Xu, Jia


The subgradient projector plays an important role in convex optimization, since the subgradient projection algorithm is a classical method for solving convex feasibility problems. This motivates us to explore fundamental properties of the subgradient projector. One can define the subgradient projector of a convex function via a selection of the subdifferential operator. This opens the door to define the subgradient projector of nonconvex functions by using the Mordukhovich limiting subdifferential operator. This thesis offers a systematic study of the subgradient projector. First, motivated by Polyak and Crombez, we present and analyze a more general algorithm for finding a fixed point of a cutter which is assumed to have a fixed point set with nonempty interior. Our results complement and extend conclusions by Crombez for cutters and by Polyak for subgradient projectors. We also present a comprehensive list of properties of the subgradient projectors which complements work by Pauwels. Special attention is given to continuity, nonexpansiveness and monotonicity. Finally, for subgradient projectors associated with nonconvex functions, we obtain various characterizations.

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