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The Lazy Travelling Salesman Problem in $\mathbb{R}^2$

Published online by Cambridge University Press:  20 June 2007

Paz Polak  and
Gershon Wolansky
Paz Polak
Affiliation:
Weizmann Institute of Science, Rehovot, Israel.
Gershon Wolansky
Affiliation:
Department of Mathematics, Technion, Haifa 32000, Israel; gershonw@math.technion.ac.il
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Abstract

We study a parameter (σ) dependent relaxation of the Travelling Salesman Problem on $\mathbb{R}^2$. The relaxed problem is reduced to the Travelling Salesman Problem as $\sigma\rightarrow$ 0. For increasing σ it is also an ordered clustering algorithm for a set of points in $\mathbb{R}^2$. A dual formulation is introduced, which reduces the problem to a convex optimization, provided the minimizer is in the domain of convexity of the relaxed functional. It is shown that this last condition is generically satisfied, provided σ is large enough.

Keywords

Travelling Salesman ProblemLegendre-Fenchel transform
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 13 , Issue 3 , July 2007 , pp. 538 - 552
Copyright
© EDP Sciences, SMAI, 2007

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