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A semidefinite optimization approach to the Target Visitation Problem

Abstract

We propose an exact algorithm for the Target Visitation Problem (TVP). The (TVP) is a composition of the Linear Ordering Problem and the Traveling Salesman Problem. It has several military and non-military applications, where two important, often competing factors are the overall distance traveled (e.g. by an unmanned aerial vehicle) and the visiting sequence of the various targets or points of interest. Hence our algorithm can be used to find the optimal visiting sequence of various pre-determined targets. First we show that the (TVP) is a special Quadratic Position Problem. Building on this finding we propose an exact semidefinite optimization approach to tackle the (TVP) and finally demonstrate its efficiency on a variety of benchmark instances with up to 50 targets.

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Notes

  1. 1.

    The Branch-and-Cut algorithm by Applegate et al. [3] holds the current record, solving an instance with 85,900 cities.

  2. 2.

    The first target is visited twice such that the tour is closed.

  3. 3.

    We note that the content of the companion paper is fairly disjoint from this paper: it deals with the (QPP) from a polyhedral point of view and applies the resulting SDP relaxations to facet defining inequalities of small (TSP) and (LOP) polytopes to facilitate the theoretical analysis of the relaxations proposed. In the companion paper we do not consider the (TVP) nor conduct any large-scale computations. Due to space restrictions we omit the proofs concerning the (QPP) and refer to our companion paper for details.

  4. 4.

    If the benefits are negative, the associated optimization problem in fact minimizes the total costs over all assignments.

  5. 5.

    We do note count the diagonal entries as the have to be \(0\) by definition.

  6. 6.

    See https://www.iwr.uni-heidelberg.de/groups/comopt/software/index.html for details.

  7. 7.

    When combining the “pal”-Instances with “br17” and “ftv33”, we multiply the binary entries by 10 and 100 respectively to balance the (LOP) benefits with the (TSP) distances. Otherwise the optimal (TVP) tour would be very similar to the optimal (TSP) tour.

  8. 8.

    The instances are available from [24].

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Correspondence to P. Hungerländer.

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Hungerländer, P. A semidefinite optimization approach to the Target Visitation Problem. Optim Lett 9, 1703–1727 (2015). https://doi.org/10.1007/s11590-014-0824-9

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Keywords

  • Target Visitation Problem
  • Linear Ordering Problem
  • Traveling Salesman Problem
  • Semidefinite Programming
  • Global Optimization