Journal of Convex Analysis 23 (2016), No. 2, 481--510
Copyright Heldermann Verlag 2016
Solving Composite Monotone Inclusions in Reflexive Banach Spaces by Constructing Best Bregman Approximations from Their Kuhn-Tucker Set
Patrick L. Combettes
Sorbonne Universités, UPMC Univ. Paris 06, Laboratoire Jacques-Louis Lions, 75005 Paris, France
plc@ljll.math.upmc.fr
Quang Van Nguyen
Sorbonne Universités, UPMC Univ. Paris 06, Laboratoire Jacques-Louis Lions, 75005 Paris, France
quangnv@ljll.math.upmc.fr
We introduce the first operator splitting method for composite monotone inclusions outside of Hilbert spaces. The proposed primal-dual method constructs iteratively the best Bregman approximation to an arbitrary point from the Kuhn-Tucker set of a composite monotone inclusion. Strong convergence is established in reflexive Banach spaces without requiring additional restrictions on the monotone operators or knowledge of the norms of the linear operators involved in the model. The monotone operators are activated via Bregman distance-based resolvent operators. The method is novel even in Euclidean spaces, where it provides an alternative to the usual proximal methods based on the standard distance.
Keywords: Best approximation, Banach space, Bregman distance, duality, Legendre function, monotone operator, operator splitting, primal-dual algorithm.
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