We consider a class of sequential network interdiction problem settings where the interdictor has incomplete initial information about the network while the evader has complete knowledge of the network including its structure and arc costs. In each decision epoch, the interdictor can block (for the duration of the epoch) at most k arcs known to him/her. By observing the evader's actions, the interdictor learns about the network structure and costs and thus, can adjust his/her actions in subsequent decision epochs. It is known from the literature that if the evader is greedy (i.e., the shortest available path is used in each decision epoch), then under some assumptions the greedy interdiction policies that block k-most vital arcs in each epoch are efficient and have a finite regret. In this paper, we consider the evader's perspective and explore deterministic "strategic" evasion policies under the assumption that the interdictor is greedy. We first study the theoretical computational complexity of the evader's problem. Then we derive basic constructive properties of optimal evasion policies for two decision epochs when the interdictor has no initial information about the network structure. These properties are then exploited for the design of a heuristic algorithm for a strategic evader in a general setting with an arbitrary time horizon and any initial information available to the interdictor. Our computational experiments demonstrate that the proposed heuristic outperforms the greedy evasion policy on several classes of synthetic network instances under either perfect or noisy information feedback. Finally, some interesting insights from our theoretical and computational results conclude the paper.
In this study we analyze linear combinatorial optimization problems where the cost vector is not known a priori, but is only observable through a finite data set. In contrast to the related studies, we presume that the number of observations with respect to particular components of the cost vector may vary. The goal is to find a procedure that transforms the data set into an estimate of the expected value of the objective function (which is referred to as a prediction rule) and a procedure that retrieves a candidate decision (which is referred to as a prescription rule). We aim at finding the least conservative prediction and prescription rules, which satisfy some specified asymptotic guarantees. We demonstrate that the resulting vector optimization problems admit a weakly optimal solution, which can be obtained by solving a particular distributionally robust optimization problem. Specifically, the decision-maker may optimize the worst-case expected loss across all probability distributions with given component-wise relative entropy distances from the empirical marginal distributions. Finally, we perform numerical experiments to analyze the out-of-sample performance of the proposed solution approach.
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 firstorder 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 maximizes the user's objective function by selecting a distribution of arc costs from the family of admissible distributions. In contrast to most of the previous studies in the related literature, both the user and the attacker can dynamically adjust their decisions at each node of the user's path. By observing the user's decisions, the attacker needs to reveal some additional distributional information associated with the arcs emanated from the current user's position. It is shown that the resulting multistage distributionally robust shortest path problem 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.
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