  We propose new primal-dual decomposition algorithms for solving systems of
inclusions involving sums of linearly composed maximally monotone operators.
The principal innovation in these algorithms is that they are block-iterative
in the sense that, at each iteration, only a subset of the monotone operators
needs to be processed, as opposed to all operators as in established methods.
Deterministic strategies are used to select the blocks of operators activated
at each iteration. In addition, we allow for operator processing "lags",
permitting asynchronous implementation. The decomposition phase of each
iteration of our methods is to generate points in the graphs of the selected
monotone operators, in order to construct a half-space containing the
Kuhn-Tucker set associated with the system. The coordination phase of each
iteration involves a projection onto this half-space. We present two related
methods: the first method provides weakly convergent primal and dual sequences
under general conditions, while the second is a variant in which strong
convergence is guaranteed without additional assumptions. Neither algorithm
requires prior knowledge of bounds on the linear operators involved or the
inversion of linear operators. Our algorithmic framework unifies and
significantly extends the approaches taken in earlier work on primal-dual
projective splitting methods.
