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
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 propos...
| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-10-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/10/21/4009 |
| Summary: | 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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| ISSN: | 2227-7390 |