## The Coolest Path Problem

Please always quote using this URN: urn:nbn:de:0297-zib-11571

- We introduce the coolest path problem, which is a mixture of two well-known problems from distinct mathematical fields. One of them is the shortest path problem from combinatorial optimization. The other is the heat conduction problem from the field of partial differential equations. Together, they make up a control problem, where some geometrical object traverses a digraph in an optimal way, with constraints on intermediate or the final state. We discuss some properties of the problem and present numerical solution techniques. We demonstrate that the problem can be formulated as a linear mixed-integer program. Numerical solutions can thus be achieved within one hour for instances with up to 70 nodes in the graph.

Author: | Martin Frank, Armin Fügenschuh, Michael Herty, Lars Schewe |
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Document Type: | ZIB-Report |

MSC-Classification: | 35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Kxx Parabolic equations and systems [See also 35Bxx, 35Dxx, 35R30, 35R35, 58J35] / 35K05 Heat equation |

90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING / 90Bxx Operations research and management science / 90B10 Network models, deterministic | |

90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING / 90Cxx Mathematical programming [See also 49Mxx, 65Kxx] / 90C10 Integer programming | |

Date of first Publication: | 2009/11/23 |

Series (Serial Number): | ZIB-Report (09-37) |

ISSN: | 1438-0064 |

Published in: | Appeared in: NETWORKS AND HETEROGENEOUS MEDIA, Volume 5, Number 1, March 2010 pp. 143–162. doi:10.3934/nhm.2010.5.143 |