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 - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Mathematics Année : 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

Résumé

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. 
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Dates et versions

hal-03880034 , version 1 (30-11-2022)

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  • HAL Id : hal-03880034 , version 1

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Chifaa Dahik, Zeina Al Masry, Stéphane Chretien, Jean-Marc Nicod, Landy Rabehasaina. 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 , 2022, 10 (21), pp.1 - 21. ⟨hal-03880034⟩
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