On the multi-stage shortest path problem under distributional uncertainty

TL;DR

This paper studies a multi-stage shortest path problem under distributional uncertainty, modeling an ambiguity-averse game between a user and an attacker, with a focus on adaptive decision-making and robust optimization.

Contribution

It introduces a novel multi-stage distributionally robust shortest path model with adaptive strategies and provides a linear mixed-integer programming reformulation for both acyclic and general graphs.

Findings

The MIP reformulation effectively captures adaptive decision-making.

Numerical results demonstrate the approach's computational tractability.

Adaptive strategies improve robustness against distributional ambiguity.

Abstract

In this paper we consider an ambiguity-averse multi-stage network game between a user and an attacker. The arc costs are assumed to be random variables that satisfy prescribed first-order moment constraints for some subsets of arcs and individual probability constraints for some particular arcs. The user aims at minimizing its cumulative expected loss by traversing between two fixed nodes in the network, while the attacker's objective is to maximize the user's expected loss by selecting a distribution of arc costs from the family of admissible distributions. In contrast to most of the related studies, both the user and the attacker can dynamically adjust their decisions at particular nodes of the user's path. By observing the user's decisions, the attacker may reveal some additional distributional information associated with the arcs emanated from the current user's position. It is shown that the resulting multi-stage distributionally robust shortest path problem (DRSPP) admits a linear mixed-integer programming reformulation (MIP). In particular, we distinguish between acyclic and general graphs by introducing different forms of non-anticipativity constraints. Finally, we perform a numerical study, where the quality of adaptive decisions and computational tractability of the proposed MIP reformulation are explored with respect to several classes of synthetic network~instances.