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Discrete Approximations to Continuum Optimal Flow Problems
Author(s) -
Lippert Ross A.
Publication year - 2006
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.1467-9590.2006.00357.x
Subject(s) - mathematics , discretization , partial differential equation , nonlinear system , mathematical optimization , duality (order theory) , nonlinear programming , quadratic growth , convex optimization , flow (mathematics) , second order cone programming , regular polygon , mathematical analysis , geometry , physics , discrete mathematics , quantum mechanics
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.