The continuous max flow problem is posed as follows :

sup $\int_\Omega p_s(x)dx$

subject to :

$|p(x)| \le C(x); \forall x \in \Omega $

$p_s(x) \le C_s(x); \forall x \in \Omega $

$p_t(x) \le C_t(x); \forall x \in \Omega $

$\nabla \cdot p(x) - p_s(x) + p_t(x) = 0; \forall x \in \Omega $

Here $p(x)$ is a field vector and is analogous to the flow in the discrete domain. $\nabla \cdot p$ is the divergence of the field p.

How do i find out the dual of this maximization problem using the lagrangian dual technique, i.e. the equivalent min cut formulation of the problem in the continuous domain.


1 Answer 1


Your question may be answered in the following paper:

@article{strang1983maximal, title={Maximal flow through a domain}, author={Strang, G.}, journal={Mathematical Programming}, volume={26}, number={2}, pages={123--143}, year={1983}, publisher={Springer} }

  • 1
    $\begingroup$ The paper does not take the approach that i want to follow. I believe that the dual can be obtained by simply finding the lagrangian function. I just did not know how to find the lagrangian dual function for the given function. I have now worked it out. $\endgroup$
    – AnkurVj
    Jun 8, 2011 at 6:14

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