<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" display="inline" overflow="scroll"><mml:mi>K</mml:mi></mml:math>-adaptability in two-stage distributionally robust binary programming

Hanasusanto, Grani A.; Kühn, Daniel; Wiesemann, Wolfram

doi:10.1016/j.orl.2015.10.006

Cited by 38 publications

(32 citation statements)

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“…It is proved that if the number of calculated solutions K is greater or equal to a certain value which depends on the dimension of the problem and other problem parameters then the K-adaptability problem calculates an exact solution. This result can be easily adapted to a distributionally robust version of Problem (M 3 ) which is the problem where P is a given set of probability distributions of the probability vector c. Similar to the result of Corollary 4.3 using the idea of the proof in [42] we can transform the latter problem into a continuous linear program if the number of calculated solutions k is greater or equal a certain value. Therefore under the latter assumption this problem can be solved in pseudopolynomial time.…”

Section: Chapter 7 Outlookmentioning

confidence: 89%

“…disutility functions (see [42]). The main task in the latter approach is to define the ambiguity set P. There are numerous different definitions of ambiguity sets in literature.…”

Section: Chapter 3 Combinatorial Robust Optimizationmentioning

confidence: 99%

“…The approach of distributional robustness received increasing attention in the last years and has been intensively studied in the literature. The K-adaptability approach which was studied in [41] for two-stage robust problems was also applied to distributionally robust problems [42]. Similar to the two-stage version the K-adaptable approach is used to approximate the distributionally robust problem.…”

Section: Chapter 7 Outlookmentioning

confidence: 99%

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Min–max–min robust combinatorial optimization

Buchheim

Kurtz

2016

Math. Program.

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In this thesis we introduce a robust optimization approach which is based on a binary min-max-min problem. The so called Min-max-min Robust Optimization extends the classical min-max approach by calculating k different solutions instead of one.Usually in robust optimization we consider problems whose problem parameters can be uncertain. The basic idea is to define an uncertainty set U which contains all relevant problem parameters, called scenarios. The objective is then to calculate a solution which is feasible for every scenario in U and which optimizes the worst objective value over all scenarios in U .As a special case of the K-adaptability approach for robust two-stage problems, the min-max-min robust optimization approach aims to calculate k different solutions for the underlying combinatorial problem, such that, considering the best of these solutions in each scenario, the worst objective value over all scenarios is optimized. This idea can be modeled as a min-max-min problem.In this thesis we analyze the complexity of the afore mentioned problem for convex and for discrete uncertainty sets U . We will show that under further assumptions the problem is as easy as the underlying combinatorial problem for convex uncertainty sets if the number of calculated solutions is greater than the dimension of the problem. Additionally we present a practical exact algorithm to solve the min-max-min problem for any combinatorial problem, given by a deterministic oracle. On the other hand we prove that if we fix the number of solutions k, then the problem is NP -hard even for polyhedral uncertainty sets and the unconstrained binary problem. For the case when the number of calculated solutions is lower or equal to the dimension we present a heuristic algorithm which is based on the exact algorithm above. Both algorithms are tested and analyzed on random instances of the knapsack problem, the vehicle routing problem and the shortest path problem.For discrete uncertainty sets we show that the min-max-min problem is NPhard for a selection of combinatorial problems. Nevertheless we prove that it can be solved in pseudopolynomial time or admits an FPTAS if the min-max problem can be solved in pseudopolynomial or admits an FPTAS respectively. ZusammenfassungIn dieser Arbeit führen wir einen Ansatz in der robusten Optimierung ein, der auf einem binären Min-max-min Problem basiert. Die sogenannte Minmax-min robuste Optimierung erweitert die klassische robuste Optimierung, indem k verschiedene Lösungen anstatt einer Einzigen berechnet werden.Im Allgemeinen betrachtet man in der robusten Optimierung Probleme, deren Problemparameter unsicher sein können. Die Grundidee ist eine Unsicherheitsmenge U zu definieren, die alle relevanten Problemparameter enthält, die man auch Szenarien nennt. Das Ziel ist es eine Lösung zu berechnen, die für alle Szenarien in U zulässig ist und die den schlechtesten Zielfunktionswert bezüglich aller Szenarien in U optimiert.Als Spezialfall des K-adaptability Ansatzes für zweistufige robuste Proble...

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Section: Chapter 7 Outlookmentioning

confidence: 89%

“…disutility functions (see [42]). The main task in the latter approach is to define the ambiguity set P. There are numerous different definitions of ambiguity sets in literature.…”

Section: Chapter 3 Combinatorial Robust Optimizationmentioning

confidence: 99%

Section: Chapter 7 Outlookmentioning

confidence: 99%

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Min–max–min robust combinatorial optimization

Buchheim

Kurtz

2016

Math. Program.

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“…In the presence of constraint uncertainty, however, these problems become intractable. Two-stage distributionally robust binary programs with polyhedral moment information are studied in [29]. If only the cost coefficients are uncertain, these problems can be reformulated as explicit mixed-integer linear programs of polynomial sizes.…”

Section: Introductionmentioning

confidence: 99%

Conic Programming Reformulations of Two-Stage Distributionally Robust Linear Programs over Wasserstein Balls

Hanasusanto

Kühn

2018

Operations Research

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153

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Adaptive robust optimization problems are usually solved approximately by restricting the adaptive decisions to simple parametric decision rules. However, the corresponding approximation error can be substantial. In this paper we show that two-stage robust and distributionally robust linear programs can often be reformulated exactly as conic programs that scale polynomially with the problem dimensions.Specifically, when the ambiguity set constitutes a 2-Wasserstein ball centered at a discrete distribution, then the distributionally robust linear program is equivalent to a copositive program (if the problem has complete recourse) or can be approximated arbitrarily closely by a sequence of copositive programs (if the problem has sufficiently expensive recourse). These results directly extend to the classical robust setting and motivate strong tractable approximations of two-stage problems based on semidefinite approximations of the copositive cone. We also demonstrate that the two-stage distributionally robust optimization problem is equivalent to a tractable linear program when the ambiguity set constitutes a 1-Wasserstein ball centered at a discrete distribution and there are no support constraints. arXiv:1609.07505v2 [math.OC]

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“…Remark 1. The K-means adaptive markdown policy described in this section is related to the K-adaptive policy presented by Bertsimas and Caramanis (2010), Hanasusanto et al (2015bHanasusanto et al ( , 2016, and Subramanyam et al (2017). Although both policies allow for discrete recourse, the clusters in K-adaptive models are determined via optimization whereas the clusters in K-means adaptive models are predetermined by clustering on data.…”

Section: K-means Adaptive Markdown Policymentioning

confidence: 99%