A Probabilistic Model for Minmax Regret in Combinatorial Optimization

Natarajan, Karthik; Shi, Dongjian; Toh, Kim-Chuan

doi:10.1287/opre.2013.1212

Cited by 18 publications

(12 citation statements)

References 50 publications

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“…Based on the data described above, the optimal bidding quantity under the minimax regret criterion, according to (10), can be calculated hourly by assessing conditions involving both of the prices and the limited information of wind power distribution. In order to investigate the effectiveness of the proposed bidding strategy, one possible way is to calculate the profit loss under a certain distribution of wind power from not making the optimal solutions.…”

Section: Resultsmentioning

confidence: 99%

“…Although the optimal wind bidding problems under the circumstance of lacking of distribution information of wind power are not fully discussed in literature, many different approaches [9][10][11][12], such as the maximin criterion, the minimax regret criterion and entropy maximization have been applied to the distribution-free newsvendor problem. Based on the mathematical equivalence of the two problems, some methods for solving the newsvendor problem may be suitable for the wind bidding problems.…”

Section: Introductionmentioning

confidence: 99%

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Wind Power Bidding Strategy Based on the Minimax Regret Criterion with Limited Distribution Information

Mao¹,

Tian²,

Zhai³

2014

JPEE

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In optimal wind bidding strategy related literatures, it is usually assumed that the full distribution information (for example, the cumulative distribution function or the probability density function) of wind power output is known. In real world applications, however, only very limited distribution information can be obtained. Therefore, the "optimal bidding strategy" obtained based on the hypothetical distribution may be far away from the true optimal one. In this paper, an optimal bidding strategy is obtained based on the minimax regret criterion. The salient feature of the new approach is that it requires only partial information of wind power distribution, for example, the expectation and the support set. Numerical test is then performed and the results suggest that the method established in this paper is effective.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Wind Power Bidding Strategy Based on the Minimax Regret Criterion with Limited Distribution Information

Mao¹,

Tian²,

Zhai³

2014

JPEE

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“…All of the works mentioned above assumed the known probability distribution, while this paper studies DRBCP-D (2b) with unknown probability distributions. As far as we are concerned, only in [30], the authors studied a generalization of DRBCP-D to minimize the worst-case sum of Γ highest costs of elements with known marginal distributions, while different from [30], this paper studies both DRBCP-U and DRBCP-D under the less conservative Wasserstein ambiguity set.…”

Section: Relevant Literaturementioning

confidence: 99%

“…where we assume that all the elements in X have the size at least Γ . Please note that in [30], the authors studied DRΓ BCP-D with known marginal distributions, while this paper studies both DRΓ BCP-U and DRΓ BCP-D under the less conservative Wasserstein ambiguity set.…”

Section: Extension: Distributionally Robust γ -Sum Bottleneck Combina...mentioning

confidence: 99%

Distributionally Robust Bottleneck Combinatorial Problems: Uncertainty Quantification and Robust Decision Making

Xie

Zhang

Ahmed

2020

Preprint

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In a bottleneck combinatorial problem, the objective is to find a subset to minimize its highest cost of elements from the combinatorial solution space. This paper studies data-driven distributionally robust bottleneck combinatorial problems (DRBCP) with stochastic costs, where the probability distribution of the cost vector is contained in a ball of distributions centered at the empirical distribution specified by the Wasserstein distance. We study two distinct versions of DRBCP from different applications: (i) Motivated by the multi-hop wireless network application, we first study the uncertainty quantification of DRBCP (denoted by DRBCP-U), where decision-makers would like to have an accurate estimation of the worst-case value of DRBCP. The difficulty of DRBCP-U is to handle its max-min-max form. Fortunately, similar to the strong duality of linear programming, the alternative forms of the bottleneck combinatorial problems from their blockers allow us to derive equivalent deterministic reformulations, which can be computed via mixed-integer programs. In addition, by drawing the connection between DRBCP-U and its sampling average approximation counterpart under empirical distribution, we show that the Wasserstein radius can be chosen in the order of negative square root of sample size, improving the existing known results; and (ii) Next, motivated by the ride-sharing application, decision-makers choose the best service-andpassenger matching that minimizes the unfairness. This gives rise to the decision-making DRBCP (denoted by DRBCP-D). For DRBCP-D, we show that its optimal solution is also optimal to its sampling average approximation counterpart, and the Wasserstein radius can be chosen in a similar order as DRBCP-U. When the sample size is small, we propose to use the optimal value of DRBCP-D to construct an indifferent solution space and propose an alternative decision-robust model, which finds the best indifferent solution to minimize the empirical variance. We further show that the decision robust model can be recast as a mixedinteger program. Finally, we extend the proposed models and solution approaches to the distributionally robust Γ −sum bottleneck combinatorial problem (DRΓ BCP), where decision-makers are interested in minimizing the worst-case sum of Γ highest costs of elements.

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“…The conditional value at risk is closely connected with the value at risk (VaR) criterion (see, e.g., [32]), which is just the α-quantile of a random outcome. Both risk criteria have attracted considerable attention in stochastic optimization (see, e.g., [29,9,30,31]). This paper is motivated by the recent papers [36] and [4], in which the following stochastic scheduling models were discussed.…”

Section: Introductionmentioning

confidence: 99%

Risk-averse single machine scheduling: complexity and approximation

Kasperski

Zieliński

2019

J Sched

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In this paper a class of single machine scheduling problems is considered. It is assumed that job processing times and due dates can be uncertain and they are specified in the form of discrete scenario set. A probability distribution in the scenario set is known. In order to choose a schedule some risk criteria such as the value at risk (VaR) an conditional value at risk (CVaR) are used. Various positive and negative complexity results are provided for basic single machine scheduling problems. In this paper new complexity results are shown and some known complexity results are strengthen. 4 d i J x 1 0 0 1 0 0 2 0 0 0 2 J x 1 1 0 0 1 1 2 0 0 0 2 J x 2 1 1 0 0 0 0 2 0 0 2 J x 2 0 0 1 1 0 0 2 0 0 2 J x 3 1 1 0 0 0 0 0 2 0 2 J x 3 0 0 0 1 1 0 0 2 0 2 J x 4 0 0 1 0 1 0 0 0 2 2 J x 4 0 1 0 0 0 0 0 0 2 2

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A Probabilistic Model for Minmax Regret in Combinatorial Optimization

Cited by 18 publications

References 50 publications

Wind Power Bidding Strategy Based on the Minimax Regret Criterion with Limited Distribution Information

Wind Power Bidding Strategy Based on the Minimax Regret Criterion with Limited Distribution Information

Distributionally Robust Bottleneck Combinatorial Problems: Uncertainty Quantification and Robust Decision Making

Risk-averse single machine scheduling: complexity and approximation

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