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

TL;DR

This paper introduces a semi-definite programming relaxation for the Robust Shortest Path Problem with ellipsoidal uncertainty, utilizing Pierra's decomposition and a Frank-Wolfe heuristic to evaluate duality gaps effectively.

Contribution

It proposes a novel SDP relaxation for the RSPP, combined with a decomposition method and a new heuristic for duality gap assessment, enhancing validation for large-scale problems.

Findings

The SDP relaxation provides a reliable lower bound.

The gap between relaxed and heuristic solutions is small.

The method is effective for large problem instances.

Abstract

This work addresses the Robust counterpart of the Shortest Path Problem (RSPP) with a correlated uncertainty set. Since this problem is hard, a heuristic approach, based on Frank-Wolfe's algorithm named Discrete Frank-Wolf (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 DFW Algorithm. The relaxed problem results from a bidualization that is done {through} a reformulation of the RSPP into a quadratic problem. Then the relaxed problem is solved 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.