Discrete Approximations to Continuum Optimal Flow Problems
Article first published online: 13 OCT 2006
Studies in Applied Mathematics
Volume 117, Issue 4, pages 321–333, November 2006
How to Cite
Lippert, R. A. (2006), Discrete Approximations to Continuum Optimal Flow Problems. Studies in Applied Mathematics, 117: 321–333. doi: 10.1111/j.1467-9590.2006.00357.x
- Issue published online: 13 OCT 2006
- Article first published online: 13 OCT 2006
- (Received May 26, 2006)
Problems in partial differential equations with inequality constraints can be used to describe a continuum analog to various optimal flow/cut problems. While general concepts from convex optimization (like duality) carry over into continuum problems, the application of ideas and algorithms from linear programming and network flow problems is challenging. The capacity constraints are nonlinear (but convex).
In this article, we investigate a discretized version of the planar maximum flow problem that preserves the nonlinear capacity constraints of the continuum problem. The resulting finite-dimensional problem can be cast as a second-order cone programming problem or a quadratically constrained program. Good numerical results can be obtained using commercial solvers. These results are in agreement with the continuum theory of a “challenge” problem posed by Strang.