Sharing the value‐at‐risk under distributional ambiguity

Chen, Zhi; Xie, Weijun

doi:10.1111/mafi.12296

Cited by 26 publications

(9 citation statements)

References 43 publications

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“…Following the similar derivation in theorem 7 [13] and according to the definition of CVaR (see, e.g., [41]), set Z q is equivalent to…”

Section: Corollary 11mentioning

confidence: 99%

ALSO-X#: Better Convex Approximations for Distributionally Robust Chance Constrained Programs

Jiang¹,

Xie²

2023

Preprint

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This paper studies distributionally robust chance constrained programs (DRCCPs), where the uncertain constraints must be satisfied with at least a probability of a prespecified threshold for all probability distributions from the Wasserstein ambiguity set. As DRCCPs are often nonconvex and challenging to solve optimally, researchers have been developing various convex inner approximations. Recently, ALSO-X has been proven to outperform the conditional value-at-risk (CVaR) approximation of a regular chance constrained program when the deterministic set is convex. In this work, we relax this assumption by introducing a new ALSO-X# method for solving DRCCPs. Namely, in the bilevel reformulations of ALSO-X and CVaR approximation, we observe that the lower-level ALSO-X is a special case of the lower-level CVaR approximation and the upper-level CVaR approximation is more restricted than the one in ALSO-X. This observation motivates us to propose the ALSO-X#, which still resembles a bilevel formulation -in the lower-level problem, we adopt the more general CVaR approximation, and for the upper-level one, we choose the less restricted ALSO-X. We show that ALSO-X# can always be better than the CVaR approximation and can outperform ALSO-X under regular chance constrained programs and type ∞−Wasserstein ambiguity set. We also provide new sufficient conditions under which ALSO-X# outputs an optimal solution to a DRCCP. We apply the proposed ALSO-X# to a wireless communication problem and numerically demonstrate that the solution quality can be even better than the exact method.

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“…Following the similar derivation in theorem 7 [13] and according to the definition of CVaR (see, e.g., [41]), set Z q is equivalent to…”

Section: Corollary 11mentioning

confidence: 99%

ALSO-X#: Better Convex Approximations for Distributionally Robust Chance Constrained Programs

Jiang¹,

Xie²

2023

Preprint

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“…(iii) Wasserstein ambiguity set is specified by the bounded distance between a nominal distribution and true distribution via Wasserstein metric (Mohajerin Esfahani and Kuhn 2017, Blanchet and Murthy 2019, Blanchet et al 2016, Chen et al 2018, Chen and Xie 2019, Gao and Kleywegt 2016, Gao et al 2017, Hanasusanto and Kuhn 2018, Bertsimas et al 2018a, Luo and Mehrotra 2017, Xie 2018, Xie and Ahmed 2019, Zhao and Guan 2018.…”

Section: Related Literaturementioning

confidence: 99%

“…and (iii) Tractability. There have been many successful developments on tractable reformulations of distributionally robust optimization with Wasserstein ambiguity set, see, for example, Mohajerin Esfahani and Kuhn (2017), Gao and Kleywegt (2016), Blanchet and Murthy (2019), Blanchet et al (2016), Gao et al (2017), Chen and Xie (2019). However, for DRTSP, the tractable results are quite limited.…”

Section: Introduction 1settingmentioning

confidence: 99%

On distributionally robust chance constrained programs with Wasserstein distance

2019

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In the optimization under uncertainty, decision-makers first select a wait-and-see policy before any realization of uncertainty and then place a here-and-now decision after the uncertainty has been observed. Two-stage stochastic programming is a popular modeling paradigm for the optimization under uncertainty that the decision-makers first specifies a probability distribution, and then seek the best decisions to jointly optimize the deterministic wait-and-see and expected here-and-now costs. In practice, such a probability distribution may not be fully available but is probably observable through an empirical dataset. Therefore, this paper studies distributionally robust two-stage stochastic program (DRTSP) which jointly optimizes the deterministic wait-and-see and worst-case expected here-and-now costs, and the probability distribution comes from a family of distributions which are centered at the empirical distribution using ∞−Wasserstein metric. There have been successful developments on deriving tractable approximations of the worst-case expected here-and-now cost in DRTSP. Unfortunately, limited results on exact tractable reformulations of DRTSP. This paper fills this gap by providing sufficient conditions under which the worst-case expected here-and-now cost in DRTSP can be efficiently computed via a tractable convex program. By exploring the properties of binary variables, the developed reformulation techniques are extended to DRTSP with binary random parameters. The main tractable reformulations in this paper are projected into the original decision space and thus can be interpreted as conventional two-stage stochastic programs under discrete support with extra penalty terms enforcing the robustness. These tractable results are further demonstrated to be sharp through complexity analysis.

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“…We remark that: (i) to obtain v U , one needs to solve N MIPs (11), which can be done in parallel; and (ii) as long as r is a rational number, using the conic quadratic representation results in [4], the MIP (11) essentially can be formulated as a mixed-integer second order conic program (MISOCP).…”

Section: Mixed-integer Programming Representationmentioning

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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Sharing the value‐at‐risk under distributional ambiguity

Cited by 26 publications

References 43 publications

ALSO-X#: Better Convex Approximations for Distributionally Robust Chance Constrained Programs

ALSO-X#: Better Convex Approximations for Distributionally Robust Chance Constrained Programs

On distributionally robust chance constrained programs with Wasserstein distance

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

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