  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.
