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Improved Bounds for the Symmetric Rendezvous Value on the Line

Author

Listed:
  • Qiaoming Han

    (School of Engineering and Management, Nanjing University, Nanjing 210093, China)

  • Donglei Du

    (Faculty of Business Administration, University of New Brunswick, Fredericton, New Brunswick, Canada E3B 5A3)

  • Juan Vera

    (Department of Management Sciences, Faculty of Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1)

  • Luis F. Zuluaga

    (Faculty of Business Administration, University of New Brunswick, Fredericton, New Brunswick, Canada E3B 5A3)

Abstract

A notorious open problem in the field of rendezvous search is to decide the rendezvous value of the symmetric rendezvous search problem on the line, when the initial distance between the two players is two. We show that the symmetric rendezvous value is within the interval (4.1520, 4.2574), which considerably improves the previous best-known result (3.9546, 4.3931). To achieve the improved bounds, we call upon results from absorbing Markov chain theory and mathematical programming theory---particularly fractional quadratic programming and semidefinite programming. Moreover, we also establish some important properties of this problem, which could be of independent interest and useful for resolving this problem completely. Finally, we conjecture that the symmetric rendezvous value is asymptotically equal to 4.25 based on our numerical calculations.

Suggested Citation

  • Qiaoming Han & Donglei Du & Juan Vera & Luis F. Zuluaga, 2008. "Improved Bounds for the Symmetric Rendezvous Value on the Line," Operations Research, INFORMS, vol. 56(3), pages 772-782, June.
    • Handle: RePEc:inm:oropre:v:56:y:2008:i:3:p:772-782
    DOI: 10.1287/opre.1070.0439
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    References listed on IDEAS

    as
    1. Wei Shi Lim & Steve Alpern & Anatole Beck, 1997. "Rendezvous Search on the Line with More Than Two Players," Operations Research, INFORMS, vol. 45(3), pages 357-364, June.
    2. Steve Alpern, 2002. "Rendezvous Search: A Personal Perspective," Operations Research, INFORMS, vol. 50(5), pages 772-795, October.
    3. Alpern, Steve & Beck, Anatole, 1997. "Rendezvous search on the line with bounded resources: expected time minimization," European Journal of Operational Research, Elsevier, vol. 101(3), pages 588-597, September.
    4. Steve Alpern & Anatole Beck, 2000. "Pure Strategy Asymmetric Rendezvous on the Line with an Unknown Initial Distance," Operations Research, INFORMS, vol. 48(3), pages 498-501, June.
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    Cited by:

    1. Leone, Pierre & Buwaya, Julia & Alpern, Steve, 2022. "Search-and-rescue rendezvous," European Journal of Operational Research, Elsevier, vol. 297(2), pages 579-591.
    2. Steve Alpern & Thomas Lidbetter, 2015. "Optimal Trade-Off Between Speed and Acuity When Searching for a Small Object," Operations Research, INFORMS, vol. 63(1), pages 122-133, February.
    3. Pierre Leone & Steve Alpern, 2022. "A Symbolic Programming Approach to the Rendezvous Search Problem," SN Operations Research Forum, Springer, vol. 3(1), pages 1-29, March.
    4. Alpern, Steven & Lidbetter, Thomas, 2015. "Optimal trade-off between speed and acuity when searching for a small object," LSE Research Online Documents on Economics 61504, London School of Economics and Political Science, LSE Library.
    5. Pierre Leone & Steve Alpern, 2018. "Rendezvous search with markers that can be dropped at chosen times," Naval Research Logistics (NRL), John Wiley & Sons, vol. 65(6-7), pages 449-461, September.

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