  Vascular networks are used across the kingdoms of life to transport fluids,
nutrients and cellular material. A popular unifying idea for understanding the
diversity and constraints of these networks is that the conduits making up the
network are organized to optimize dissipation or other functions within the
network. However the general principles governing the optimal networks remain
unknown. In particular Durand showed that under Neumann boundary conditions
networks, that minimize dissipation should be trees. Yet many real transport
networks, including capillary beds, are not simply connected. Previously
multiconnectedness in a network has been assumed to provide evidence that the
network is not simply minimizing dissipation. Here we show that if the boundary
conditions on the flows within the network are enlarged to include physical
reasonable Neumann and Dirichlet boundary conditions (i.e. constraints on
either pressure or flow) then minimally dissipative networks need not be trees.
To get to this result we show that two methods of producing optimal networks,
namely enforcing constraints via Lagrange multipliers or via penalty methods,
are equivalent for tree networks.

Optimization and Control