Identifying effective scenarios in distributionally robust stochastic programs with total variation distance

Rahimian, Hamed; Bayraksan, Güzi̇n; Homem-de-Mello, Tito

doi:10.1007/s10107-017-1224-6

Cited by 46 publications

(28 citation statements)

References 55 publications

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“…This method is referred to as the Sample Average Approximation (SAA) and in [5], we have the convergence results regarding a few sampling methods. We find the recent advance of SAA in [6] and [12] with references therein.…”

Section: Introductionmentioning

confidence: 73%

Identifying Effective Scenarios for Sample Average Approximation

Chen¹

2019

Preprint

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We introduce a method to improve the tractability of the well-known Sample Average Approximation (SAA) without compromising important theoretical properties, such as convergence in probability and the consistency of an independent and identically distributed (iid) sample. We consider each scenario as a polyhedron of the mix of first-stage and second-stage decision variables.According to John's theorem, the Löwner-John ellipsoid of each polyhedron will be unique which means that different scenarios will have correspondingly different Löwner-John ellipsoids. By optimizing the objective function regarding both feasible regions of the polyhedron and its unique Löwner-John ellipsoid, respectively, we obtain a pair of optimal values, which would be a coordinate on a two-dimensional plane. The scenarios, whose coordinates are close enough on the plane, will be treated as one scenario; thus our method reduces the sample size of an iid sample considerably.Instead of using a large iid sample directly, we would use the cluster of low-cost computers to calculate the coordinates of a massive number of scenarios and build a representative and significantly smaller sample to feed the solver. We show that our method will obtain the optimal solution of a very large sample without compromising the solution quality. Furthermore, our method would be implementable as a distributed computational infrastructure with many but low-cost computers.

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Section: Introductionmentioning

confidence: 73%

Identifying Effective Scenarios for Sample Average Approximation

Chen¹

2019

Preprint

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“…The ess sup ξ∈Ξ f (x, ξ) represents the almost surely worst-case cost f (x, ξ) over set Ξ with regard to the Lebesgue measure. Then we give the reformulation result [93,137,155] of sup P ∈P DT V {E P [f (x, ξ)]}.…”

Section: Total-variation-metric-based Ambiguity Setsmentioning

confidence: 99%

“…Their algorithm is guaranteed to terminate in finite number of iterations and to return a ρ-optimal solution. Rahimian et al [137] considered a convex DRO model with the total-variation-distance-based ambiguity set. They proposed a cutting plane method called Primal Decomposition to solve the problems.…”

Section: Bayesian Ambiguity Setsmentioning

confidence: 99%

Distributionally Robust Optimization: A review on theory and applications

Lin¹,

Fang²,

Gao³

2022

NACO

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<p style='text-indent:20px;'>In this paper, we survey the primary research on the theory and applications of distributionally robust optimization (DRO). We start with reviewing the modeling power and computational attractiveness of DRO approaches, induced by the ambiguity sets structure and tractable robust counterpart reformulations. Next, we summarize the efficient solution methods, out-of-sample performance guarantee, and convergence analysis. Then, we illustrate some applications of DRO in machine learning and operations research, and finally, we discuss the future research directions.</p>

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“…A noteworthy example of a metric that does not satisfy the above requirement is the discrete metric, defined as d(ξ , ξ ) = 1 whenever ξ = ξ and 0 otherwise. In this case, the Wasserstein distance is equivalent to the total variation distance, and the distributions that solve the inner supremum in (2) can assign positive probability only to the training samples and the worst-case realization [29,43]. In a network with rare failures, this means that only past observed realizations and the realization corresponding to complete network failure are taken into account, resulting in poor out-of-sample performance.…”

Section: Advantages Of Wasserstein Ambiguity Sets For Discrete Rare E...mentioning

confidence: 99%

Data-driven two-stage conic optimization with zero-one uncertainties

Subramanyam¹,

Tonbari²,

Kim³

2020

Preprint

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We address high-dimensional zero-one random parameters in two-stage convex conic optimization problems. Such parameters typically represent failures of network elements and constitute rare, high-impact random events in several applications. Given a sparse training dataset of the parameters, we motivate and study a distributionally robust formulation of the problem using a Wasserstein ambiguity set centered at the empirical distribution. We present a simple, tractable, and conservative approximation of this problem that can be efficiently computed and iteratively improved. Our method relies on a reformulation that optimizes over the convex hull of a mixed-integer conic programming representable set, followed by an approximation of this convex hull using lift-and-project techniques. We illustrate the practical viability and strong out-of-sample performance of our method on nonlinear optimal power flow problems affected by random contingencies, and we report improvements of up to 20% over existing methods. MotivationThis work is motivated by optimization problems arising in applications that are affected by an extremely large, yet finite, number of rare, high-impact random events. In particular, we are motivated by applications in which the decision-relevant random events consist of high-dimensional binary outcomes. Such applications are ubiquitous in network optimization, where the uncertain 1

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Identifying effective scenarios in distributionally robust stochastic programs with total variation distance

Cited by 46 publications

References 55 publications

Identifying Effective Scenarios for Sample Average Approximation

Identifying Effective Scenarios for Sample Average Approximation

Distributionally Robust Optimization: A review on theory and applications

Data-driven two-stage conic optimization with zero-one uncertainties

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