Variable-sized uncertainty and inverse problems in robust optimization

Chassein, André; Goerigk, Marc

doi:10.1016/j.ejor.2017.06.042

Cited by 16 publications

(12 citation statements)

References 25 publications

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“…We briefly sketch approaches to generate U in the following. Each is equipped with a scaling parameter to control its size (see also [14] on the problem of choosing the size of an uncertainty set with a given shape). A visual example using four data points in two dimensions is provided for each apprach in Figure 1.…”

Section: Uncertainty Sets For the Shortest Path Problemmentioning

confidence: 99%

Algorithms and uncertainty sets for data-driven robust shortest path problems

Chassein

Dokka

Goerigk

2019

European Journal of Operational Research

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We consider robust shortest path problems, where the aim is to find a path that optimizes the worst-case performance over an uncertainty set containing all relevant scenarios for arc costs. The usual approach for such problems is to assume this uncertainty set given by an expert who can advise on the shape and size of the set.Following the idea of data-driven robust optimization, we instead construct a range of uncertainty sets from the current literature based on real-world traffic measurements provided by the City of Chicago. We then compare the performance of the resulting robust paths within and outside the sample, which allows us to draw conclusions what the most suited uncertainty set is.Based on our experiments, we then focus on ellipsoidal uncertainty sets, and develop a new solution algorithm that significantly outperforms a stateof-the-art solver.robust shortest path problems, on the other hand, are NP-hard (see [20]), and real-time information has not been an option.To formulate a robust problem, it is necessary to have a description of all possible and relevant scenarios that the solution should prepare against, the so-called uncertainty set. We refer to the surveys [1,16,17] for a general overview on the topic. The current literature on robust shortest paths usually assumes this set to be given, by some mixture of data-preprocessing and expert knowledge that is not part of the study. This means that different types of sets have been studied (compare, e.g., [18,9]), but it has been impossible to address the question which would be the "right" choice.A recent paradigm shift is data-driven robust optimization (see [6]), where building the uncertainty set from raw observations is part of the robust optimization problem. This paper is the first to follow such a perspective for shortest path problems. Based on real-world observations from the City of Chicago, we build a range of uncertainty sets, calculate the corresponding robust solutions, and perform an in-depth analysis of their performance. This allows us to give an indication which set is actually suitable for our application, and which are not.In the second part of this paper, we then focus on the case of ellipsoidal uncertainty, and provide a branch-and-bound algorithm that is able to solve instances considerably faster than an off-the-shelf solver.Parts of this paper were previously published as a conference paper in [15]. In comparison, we provide a completely new set of experimental results based on an observation period of 46 days (instead of one single day), which leads to a more detailed insight into the performance of different uncertainty sets. Furthermore, we provide a new analysis for axis-parallel ellipsoidal uncertainty sets, including an efficient branch-and-bound algorithm that is able to outperform Cplex by several orders of magnitude, pushing robust shortest paths towards applicability in real-time navigation systems.The remainder of the paper is structured as follows. In Section 2 we briefly introduce the robust shortest path probl...

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Section: Uncertainty Sets For the Shortest Path Problemmentioning

confidence: 99%

Algorithms and uncertainty sets for data-driven robust shortest path problems

Chassein

Dokka

Goerigk

2019

European Journal of Operational Research

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“…We briefly summarize the setting of [CG16b]. Consider an uncertain combinatorial problem of the form…”

Section: Variable-sized Uncertaintymentioning

confidence: 99%

“…The basic idea of variable-sized uncertainty was recently introduced in [CG16b]. There, the aim is to construct a set of robust candidate solutions that requires the decision maker to chose one that suits him best.…”

Section: Introductionmentioning

confidence: 99%

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Compromise solutions for robust combinatorial optimization with variable-sized uncertainty

Chassein

Goerigk

2018

European Journal of Operational Research

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In classic robust optimization, it is assumed that a set of possible parameter realizations, the uncertainty set, is modeled in a previous step and part of the input. As recent work has shown, finding the most suitable uncertainty set is in itself already a difficult task. We consider robust problems where the uncertainty set is not completely defined. Only the shape is known, but not its size. Such a setting is known as variable-sized uncertainty.In this work we present an approach how to find a single robust solution, that performs well on average over all possible uncertainty set sizes. We demonstrate that this approach can be solved efficiently for min-max robust optimization, but is more involved in the case of min-max regret, where positive and negative complexity results for the selection problem, the minimum spanning tree problem, and the shortest path problem are provided. We introduce an iterative solution procedure, and evaluate its performance in an experimental comparison.

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“…In [CG18c,CG18a], the authors considered a setting in which the shape of the uncertainty set is given, but not its size. Models are introduced by which compromise robust solutions can be found, which perform well on average over all considered uncertainty sizes.…”

Section: Introductionmentioning

confidence: 99%

Mixed uncertainty sets for robust combinatorial optimization

2019

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In robust optimization, the uncertainty set is used to model all possible outcomes of uncertain parameters. In the classic setting, one assumes that this set is provided by the decision maker based on the data available to her. Only recently it has been recognized that the process of building useful uncertainty sets is in itself a challenging task that requires mathematical support.In this paper, we propose an approach to go beyond the classic setting, by assuming multiple uncertainty sets to be prepared, each with a weight showing the degree of belief that the set is a "true" model of uncertainty. We consider theoretical aspects of this approach and show that it is as easy to model as the classic setting. In an extensive computational study using a shortest path problem based on real-world data, we auto-tune uncertainty sets to the available data, and show that with regard to out-sample performance, the combination of multiple sets can give better results than each set on its own.

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Variable-sized uncertainty and inverse problems in robust optimization

Cited by 16 publications

References 25 publications

Algorithms and uncertainty sets for data-driven robust shortest path problems

Algorithms and uncertainty sets for data-driven robust shortest path problems

Compromise solutions for robust combinatorial optimization with variable-sized uncertainty

Mixed uncertainty sets for robust combinatorial optimization

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