  Safe and economic operation of networked systems is often challenging.
Optimization-based schemes are frequently considered, since they achieve
near-optimality while ensuring safety via the explicit consideration of
constraints. In applications, these schemes, however, often require solving
large-scale optimization problems. Iterative techniques from distributed
optimization are frequently proposed for complexity reduction. Yet, they
achieve feasibility only asymptotically, which induces a substantial
computational burden. This work presents an approximate dynamic programming
scheme, which is guaranteed to deliver a feasible solution in "one shot", i.e.,
in one backward-forward iteration over all subproblems provided they are
coupled by a tree structure. Our proposed scheme generalizes methods from
seemingly disconnected domains such as power systems and optimal control. We
demonstrate its efficacy for problems with nonconvex constraints via numerical
examples from both domains.
