The coolest path problem

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. Continuous and discrete optimization are at present two distinct areas of mathematics. From time to time, discrete optimizers stumble over a problem which has some intrinsic nonlinear continuous structure, sometimes modeled using partial differential equations. Then they most...

• Type: Journal
• Year: 2010
• Where: NHM
• Authors: Martin Frank, Armin Fügenschuh, Michael Herty, Lars Schewe
• Keywords: NHM 2010, Numerical Solutions, Path Problem, Shortest Path Problem