An approach to the distributionally robust shortest path problem

Ketkov, Sergey S.; Prokopyev, Oleg A.; Burashnikov, Evgenii P.

doi:10.1016/j.cor.2021.105212

Cited by 8 publications

(16 citation statements)

References 46 publications

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“…In the recent Steiner bi-objective Shortest Path Problem introduced in [14], the authors present this new variant of the SPP capable of preprocessing data to solve the well-known vehicle routing problem. SPP in which the cost of the arcs is not known in advance has been studied in the recent literature [15,16]. Stochastic shortest path (SSP) dealing with applications in routing problems and in road networks can be found in [17,18].…”

Section: Literature Reviewmentioning

confidence: 99%

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The Cost-Balanced Path Problem: A Mathematical Formulation and Complexity Analysis

Ambrosino

Cerrone

2022

Mathematics

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This paper introduces a new variant of the Shortest Path Problem (SPP) called the Cost-Balanced Path Problem (CBPP). Various real problems can either be modeled as BCPP or include BCPP as a sub-problem. We prove several properties related to the complexity of the CBPP problem. In particular, we demonstrate that the problem is NP-hard in its general version, but it becomes solvable in polynomial time in a specific family of instances. Moreover, a mathematical formulation of the CBPP, as a mixed-integer programming model, is proposed, and some additional constraints for modeling real requirements are given. This paper validates the proposed model and its extensions with experimental tests based on random instances. The analysis of the results of the computational experiments shows that the proposed model and its extension can be used to model many real applications. Obviously, due to the problem complexity, the main limitation of the proposed approach is related to the size of the instances. A heuristic solution approach should be required for larger-sized and more complex instances.

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Section: Literature Reviewmentioning

confidence: 99%

“…Thanks to Equations ( 13) and ( 14), the flow ( f ij ) on each selected arc must be less or equal to its maximum value and greater or equal to the minimum required, while thanks to Equation (15) the outflow from the origin node ( f oj ) is fixed equal to the starting level (a o ).…”

Section: Model Extensionsmentioning

confidence: 99%

The Cost-Balanced Path Problem: A Mathematical Formulation and Complexity Analysis

Ambrosino

Cerrone

2022

Mathematics

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“…One natural limitation of our approach is that we exploit the decision-maker's expected loss as the objective criterion in (1), i.e., γ(c, x) = c x. First, we note that routing decisions with the least expected cost are used in intelligent transportation and in-vehicle route guidance systems [16]; we also refer to [17] for some motivation behind the expected loss criterion in a data-driven framework.…”

Section: Our Approach and Contributionmentioning

confidence: 99%

“…For example, Anderson and Philpott [23] analyze different mechanisms that hedge against uncertainty in a more sophisticated way, e.g., a CVaR-based risk measure, phi-divergence using total variation and a Wasserstein metric. At the same time, we are not aware about DRO formulations (except for a formulation in [17]) that can address incomplete data sets.…”

Section: Benchmark Approachesmentioning

confidence: 99%

On a class of data-driven mixed-integer programming problems under uncertainty: a distributionally robust approach

Ketkov¹,

Shilov²

2021

Preprint

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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.

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“…In fact, our multi-stage formulation is motivated and built upon the related one-stage formulation of DRSPP in [27]. Specifically, we preserve the same form of ambiguity set and the loss function, but introduce some auxiliary distributional constraints that can be verified by the user dynamically while traversing through the network.…”

Section: Introductionmentioning

confidence: 99%

On the multi-stage shortest path problem under distributional uncertainty

Ketkov¹

2022

Preprint

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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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An approach to the distributionally robust shortest path problem

Cited by 8 publications

References 46 publications

The Cost-Balanced Path Problem: A Mathematical Formulation and Complexity Analysis

The Cost-Balanced Path Problem: A Mathematical Formulation and Complexity Analysis

On a class of data-driven mixed-integer programming problems under uncertainty: a distributionally robust approach

On the multi-stage shortest path problem under distributional uncertainty

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