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.