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An SDP Dual Relaxation for the Robust Shortest-Path Problem with Ellipsoidal Uncertainty: Pierra’s Decomposition Method and a New Primal Frank–Wolfe-Type Heuristics for Duality Gap Evaluation

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  • Chifaa Al Dahik

    (FEMTO-ST Institute, University Bourgogne Franche-Comté, CNRS, ENSMM, 25000 Besançon, France
    Laboratoire de Mathématiques de Besançon, University Bourgogne Franche-Comté, CNRS, 25000 Besançon, France)

  • Zeina Al Masry

    (FEMTO-ST Institute, University Bourgogne Franche-Comté, CNRS, ENSMM, 25000 Besançon, France)

  • Stéphane Chrétien

    (Laboratoire ERIC, UFR ASSP, Université Lyon 2, 69500 Lyon, France)

  • Jean-Marc Nicod

    (FEMTO-ST Institute, University Bourgogne Franche-Comté, CNRS, ENSMM, 25000 Besançon, France)

  • Landy Rabehasaina

    (Laboratoire de Mathématiques de Besançon, University Bourgogne Franche-Comté, CNRS, 25000 Besançon, France)

Abstract

This work addresses the robust counterpart of the shortest path problem (RSPP) with a correlated uncertainty set. Because this problem is difficult, a heuristic approach, based on Frank–Wolfe’s algorithm named discrete Frank–Wolfe (DFW), has recently been proposed. The aim of this paper is to propose a semi-definite programming relaxation for the RSPP that provides a lower bound to validate approaches such as the DFW algorithm. The relaxed problem is a semi-definite programming (SDP) problem that results from a bidualization that is done through a reformulation of the RSPP into a quadratic problem. Then, the relaxed problem is solved by using a sparse version of Pierra’s decomposition in a product space method. This validation method is suitable for large-size problems. The numerical experiments show that the gap between the solutions obtained with the relaxed and the heuristic approaches is relatively small.

Suggested Citation

  • Chifaa Al Dahik & Zeina Al Masry & Stéphane Chrétien & Jean-Marc Nicod & Landy Rabehasaina, 2022. "An SDP Dual Relaxation for the Robust Shortest-Path Problem with Ellipsoidal Uncertainty: Pierra’s Decomposition Method and a New Primal Frank–Wolfe-Type Heuristics for Duality Gap Evaluation," Mathematics, MDPI, vol. 10(21), pages 1-21, October.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:21:p:4009-:d:956544
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    References listed on IDEAS

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    1. Christoph Buchheim & Jannis Kurtz, 2018. "Robust combinatorial optimization under convex and discrete cost uncertainty," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 6(3), pages 211-238, September.
    2. Marguerite Frank & Philip Wolfe, 1956. "An algorithm for quadratic programming," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 3(1‐2), pages 95-110, March.
    3. Baron, Opher & Berman, Oded & Fazel-Zarandi, Mohammad M. & Roshanaei, Vahid, 2019. "Almost Robust Discrete Optimization," European Journal of Operational Research, Elsevier, vol. 276(2), pages 451-465.
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