Distributionally robust optimization with matrix moment constraints: Lagrange duality and cutting plane methods

Xu, Huifu; Liu, Yongchao; Sun, Hailin

doi:10.1007/s10107-017-1143-6

Cited by 68 publications

(89 citation statements)

References 49 publications

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“…Since Ξ = IR k and the set {E P [ξ] : P ∈ P(Ξ)} = IR k , there exists a probability distribution P 0 such that E P0 [ξ] = µ, E P0 [|ξ − µ|] < d, which means the Slater condition holds; see [45]. For any Q ∈ P(Ξ), it follows from Hoffman's Lemma [42,Lemma 2] that there exists a positive constant C 1 such that…”

Section: Parts (Ii)mentioning

confidence: 99%

Convergence Analysis for Mathematical Programs with Distributionally Robust Chance Constraint

Guo¹,

Xu²,

Zhang³

2017

SIAM J. Optim.

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Abstract. Convergence analysis for optimization problems with chance constraints concerns impact of variation of probability measure in the chance constraints on the optimal value and the optimal solutions and research on this topic has been well documented in the literature of stochastic programming. In this paper, we extend such analysis to optimization problems with distributionally robust chance constraints where the true probability distribution is unknown, but it is possible to construct an ambiguity set of probability distributions and the chance constraint is based on the most conservative selection of probability distribution from the ambiguity set. The convergence analysis focuses on impact of the variation of the ambiguity set on the optimal value and the optimal solutions. We start by deriving general convergence results under abstract conditions such as continuity of the robust probability function and uniform convergence of the robust probability functions and followed with detailed analysis of these conditions. Two sufficient conditions have been derived with one applicable to both continuous and discrete probability distribution and the other to continuous distribution. Case studies are carried out for ambiguity sets being constructed through moments and samples.

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Section: Parts (Ii)mentioning

confidence: 99%

Convergence Analysis for Mathematical Programs with Distributionally Robust Chance Constraint

Guo¹,

Xu²,

Zhang³

2017

SIAM J. Optim.

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“…, N . For fixed t ∈ T N , the inner maximization in P can be formulated as a LP: 29) where ∆ N := {p ∈ IR N + : 30) or equivalently,…”

Section: Methodsmentioning

confidence: 99%

“…Condition (b) means any point in Ξ may be approximated by a point in Ξ N when N is sufficiently large. The approximation scheme (using P N to approximate P) is considered by Xu, Liu and Sun [30], where they propose a cutting-plane method for solving a minimax distributionally robust optimization problem directly. However, they are short of stating convergence of P N to P explicitly.…”

Section: Discrete Approximation Of the Ambiguity Setmentioning

confidence: 99%

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Probability approximation schemes for stochastic programs with distributionally robust second-order dominance constraints

Guo

Zhang

2016

Optimization Methods and Software

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Abstract. Since the pioneering work [7] by Dentcheva and Ruszczyński, stochastic programs with second order dominance constraints (SPSODC) have received extensive discussions over the past decade from theory of optimality to numerical schemes and practical applications. In this paper, we investigate discrete approximation of SPSODC when (a) the true probability is known but continuously distributed and (b) the true probability distribution is unknown but it lies within an ambiguity set of distributions. Differing from the well-known Monte Carlo discretization method, we propose a deterministic discrete approximation scheme due to Pflug and Pichler [20] and demonstrate that the discrete probability measure and the ambiguity set of discrete probability measures approximate their continuous counterparts under the Kantorovich metric. Stability analysis of the optimal value and optimal solutions of the resulting discrete optimization problems is presented and some comparative numerical test results are reported.

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“…In the literature of distributionally robust optimization, various statistical methods have been proposed to build ambiguity sets based on available information of the underlying uncertainty, see for instance [27,28] and the references therein. Here we consider φ-divergence ball and Kantorovich ball approaches and discuss tractable formulations of the corresponding (DRSRP').…”

Section: Structure Of (Drsrp') and Approximation Of The Ambiguity Setmentioning

confidence: 99%

Distributionally robust shortfall risk optimization model and its approximation

Guo

2018

Math. Program.

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Utility-based shortfall risk measures (SR) have received increasing attention over the past few years for their potential to quantify the risk of large tail losses more effectively than conditional value at risk. In this paper, we consider a distributionally robust version of the shortfall risk measure (DRSR) where the true probability distribution is unknown and the worst distribution from an ambiguity set of distributions is used to calculate the SR. We start by showing that the DRSR is a convex risk measure and under some special circumstance a coherent risk measure. We then move on to study an optimization problem with the objective of minimizing the DRSR of a random function and investigate numerical tractability of the optimization problem with the ambiguity set being constructed through φ-divergence ball and Kantorovich ball. In the case when the nominal distribution in the balls is an empirical distribution constructed through iid samples, we quantify convergence of the ambiguity sets to the true probability distribution as the sample size increases under the Kantorovich metric and consequently the optimal values of the corresponding DRSR problems. Specifically, we show that the error of the optimal value is linearly bounded by the error of each of the approximate ambiguity sets and subsequently derive a confidence interval of the optimal value under each of the approximation schemes. Some prelimThe research is supported by EPSRC Grant EP/M003191/1. The work of the first author was partially carried out while she was working as a postdoctoral research fellow in the School of Mathematics, University of Southampton supported by the EPSRC grant.

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Distributionally robust optimization with matrix moment constraints: Lagrange duality and cutting plane methods

Cited by 68 publications

References 49 publications

Convergence Analysis for Mathematical Programs with Distributionally Robust Chance Constraint

Convergence Analysis for Mathematical Programs with Distributionally Robust Chance Constraint

Probability approximation schemes for stochastic programs with distributionally robust second-order dominance constraints

Distributionally robust shortfall risk optimization model and its approximation

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