Robust Solutions of Optimization Problems Affected by Uncertain Probabilities

Ben‐Tal, Aharon; Hertog, Dick den; Waegenaere, Anja De; Melenberg, Bertrand; Rennen, G.

doi:10.1287/mnsc.1120.1641

Cited by 537 publications

(187 citation statements)

References 39 publications

(74 reference statements)

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“…For general φ-divergence and other examples, interested readers are referred to Sect. 2, Pardo [34], and Ben-Tal et al [4]. Based on φ-divergence, the decision makers can build a confidence set (cf.…”

Section: Model Settings and Confidence Setsmentioning

confidence: 99%

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Data-driven chance constrained stochastic program

2015

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In this paper, we study data-driven chance constrained stochastic programs, or more specifically, stochastic programs with distributionally robust chance constraints (DCCs) in a data-driven setting to provide robust solutions for the classical chance constrained stochastic program facing ambiguous probability distributions of random parameters. We consider a family of density-based confidence sets based on a general φ-divergence measure, and formulate DCC from the perspective of robust feasibility by allowing the ambiguous distribution to run adversely within its confidence set. We derive an equivalent reformulation for DCC and show that it is equivalent to a classical chance constraint with a perturbed risk level. We also show how to evaluate the perturbed risk level by using a bisection line search algorithm for general φ-divergence measures. In several special cases, our results can be strengthened such that we can derive closed-form expressions for the perturbed risk levels. In addition, we show that the conservatism of DCC vanishes as the size of historical data goes to infinity. Furthermore, we analyze the relationship between the conservatism of DCC An early version of this paper is available online at and the size of historical data, which can help indicate the value of data. Finally, we conduct extensive computational experiments to test the performance of the proposed DCC model and compare various φ-divergence measures based on a capacitated lotsizing problem with a quality-of-service requirement.

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Section: Model Settings and Confidence Setsmentioning

confidence: 99%

“…Based on φ-divergence, the decision makers can build a confidence set (cf. Ben-Tal et al [4]) as follows:…”

Section: Model Settings and Confidence Setsmentioning

confidence: 99%

Data-driven chance constrained stochastic program

2015

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“…The first is input uncertainty quantification, which quantifies the impact of input uncertainty on the simulation output (Barton, Nelson, & Xie, 2014;Song, Nelson, & Pegden, 2014). The other is robust optimization (RO) (Ben-Tal, den Hertog, Waegenaere, Melenberg, & Rennen, 2013;Delage & Ye, 2010). Different from simulationbased optimization in which targeted problems do not have nice structures to be exploited, RO often requires the optimization problems to be available explicitly in closed form.…”

Section: Introductionmentioning

confidence: 99%

Robust ranking and selection with optimal computing budget allocation

et al. 2017

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a b s t r a c tIn this paper, we consider the ranking and selection (R&S) problem with input uncertainty. It seeks to maximize the probability of correct selection (PCS) for the best design under a fixed simulation budget, where each design is measured by their worst-case performance. To simplify the complexity of PCS, we develop an approximated probability measure and derive an asymptotically optimal solution of the resulting problem. An efficient selection procedure is then designed within the optimal computing budget allocation (OCBA) framework. More importantly, we provide some useful insights on characterizing an efficient robust selection rule and how it can be achieved by adjusting the simulation budgets allocated to each scenario.

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“…This approach, however, tends to find solutions that are too conservative to provide much more optimality. Later, Ben-Tal et al [21][22][23] carry out further research on the robust optimization theory and have made significant progress in robust convex optimization. However, as the resulting robust counterparts involve nonlinear problems, such methods cannot be applied to discrete optimization.…”

Section: Related Workmentioning

confidence: 99%

Sensor Network Deployment under Distance Uncertainty with Robust Optimization

Qiao¹,

Liu²,

Zhang³

et al. 2015

Int. J. Onl. Eng.

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Abstract-We consider the sensor deployment problem in the context of distance uncertainty. It is characterized by differentiated arrangement of specific detection probability thresholds at different locations. The problem is formulated as an integer linear programming (ILP) model firstly, aiming at optimizing the number of sensors and their locations. Based on the robust discrete optimization methodology, the uncertain model is transformed into an equivalent ILP problem considering distance uncertainty. The proposed approach can control the tradeoff between optimality and robustness by varying the parameters named protection levels. Uniform and non-uniform event detection probabiliy distributions are considered in the experiment. The results show that, as the distance uncertainty increases, the constraint violation can be avoided in the robust model and the robust solution can provide a significant improvement at the expense of a small loss in optimality when compared to the optimal solution of a deterministic scenario.

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Robust Solutions of Optimization Problems Affected by Uncertain Probabilities

Cited by 537 publications

References 39 publications

Data-driven chance constrained stochastic program

Data-driven chance constrained stochastic program

Robust ranking and selection with optimal computing budget allocation

Sensor Network Deployment under Distance Uncertainty with Robust Optimization

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