The Discrete Moment Problem with Nonconvex Shape Constraints

Chen, Xi; He, Simai; Jiang, Bo; Ryan, Christopher; Zhang, Teng

doi:10.1287/opre.2020.1990

Cited by 16 publications

(6 citation statements)

References 53 publications

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“…However, there are some related problems that our framework cannot handle. These problems include non-convex-shaped moment problems (Chen et al 2021) and discrete moment problems (Ninh and Prékopa 2013, Prékopa et al 2016, Ninh et al 2019. This is due to their inherent nonconvex natures (see, e.g., Chen et al 2021) and the nonsmoothness of the functions in the dual constraints (see, e.g., Prékopa et al 2016), which we leave for future work.…”

Section: Discussionmentioning

confidence: 99%

A Unified Framework for Generalized Moment Problems: a Novel Primal-Dual Approach

Guo¹,

He²,

Jiang³

et al. 2022

Preprint

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Generalized moment problems optimize functional expectation over a class of distributions with generalized moment constraints, i.e., the function in the moment can be any measurable function. These problems have recently attracted growing interest due to their great flexibility in representing nonstandard moment constraints, such as geometry-mean constraints, entropy constraints, and exponential-type moment constraints. Despite the increasing research interest, analytical solutions are mostly missing for these problems, and researchers have to settle for nontight bounds or numerical approaches that are either suboptimal or only applicable to some special cases. In addition, the techniques used to develop closed-form solutions to the standard moment problems are tailored for specific problem structures. In this paper, we propose a framework that provides a unified treatment for any moment problem. The key ingredient of the framework is a novel primal-dual optimality condition. This optimality condition enables us to reduce the original infinite dimensional problem to a nonlinear equation system with a finite number of variables. In solving three specific moment problems, the framework demonstrates a clear path for identifying the analytical solution if one is available, otherwise, it produces semi-analytical solutions that lead to efficient numerical algorithms. Finally, through numerical experiments, we provide further evidence regarding the performance of the resulting algorithms by solving a moment problem and a distributionally robust newsvendor problem.

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Section: Discussionmentioning

confidence: 99%

A Unified Framework for Generalized Moment Problems: a Novel Primal-Dual Approach

Guo¹,

He²,

Jiang³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…In the DRO literature, the choice of U λ can be categorized roughly into two groups. The first group is based on partial distributional information, such as moment and support (Ghaoui et al 2003, Delage and Ye 2010, Goh and Sim 2010, Wiesemann et al 2014, Hanasusanto et al 2015, shape (Popescu 2005, Van Parys et al 2016, Li et al 2017, Lam and Mottet 2017, Chen et al 2021) and marginal distribution , Doan et al 2015, Dhara et al 2021. This approach has proven useful in robustifying decisions when facing limited distributional information, or when data is scarce, e.g., in the extremal region.…”

Section: Distance-based Dromentioning

confidence: 99%

On the Impossibility of Statistically Improving Empirical Optimization: A Second-Order Stochastic Dominance Perspective

Lam

2021

Preprint

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When the underlying probability distribution in a stochastic optimization is observed only through data, various data-driven formulations have been studied to obtain approximate optimal solutions. We show that no such formulations can, in a sense, theoretically improve the statistical quality of the solution obtained from empirical optimization. We argue this by proving that the first-order behavior of the optimality gap against the oracle best solution, which includes both the bias and variance, for any data-driven solution is second-order stochastically dominated by empirical optimization, as long as suitable smoothness holds with respect to the underlying distribution. We demonstrate this impossibility of improvement in a range of examples including regularized optimization, distributionally robust optimization, parametric optimization and Bayesian generalizations. We also discuss the connections of our results to semiparametric statistical inference and other perspectives in the data-driven optimization literature.

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“…The construction can be roughly categorized into two approaches: 1) neighborhood ball using statistical distance, which include most commonly φ-divergence (Ben-Tal et al 2013, Bayraksan and Love 2015, Jiang and Guan 2016, Lam 2016 and Wasserstein distance (Esfahani and Kuhn 2018, Blanchet and Murthy 2019, Gao and Kleywegt 2016, Chen and Paschalidis 2018. 2) partial distributional information including moment (Ghaoui et al 2003, Delage and Ye 2010, Goh and Sim 2010, Wiesemann et al 2014, Hanasusanto et al 2015, distributional shape (Popescu 2005, Van Parys et al 2016, Li et al 2017, Chen et al 2020) and marginal , Doan et al 2015, Dhara et al 2021 constraints.…”

Section: Related Work and Comparisonsmentioning

confidence: 99%

Generalization Bounds with Minimal Dependency on Hypothesis Class via Distributionally Robust Optimization

Zeng¹,

Lam²

2021

Preprint

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Established approaches to obtain generalization bounds in data-driven optimization and machine learning mostly build on solutions from empirical risk minimization (ERM), which depend crucially on the functional complexity of the hypothesis class. In this paper, we present an alternate route to obtain these bounds on the solution from distributionally robust optimization (DRO), a recent data-driven optimization framework based on worst-case analysis and the notion of ambiguity set to capture statistical uncertainty. In contrast to the hypothesis class complexity in ERM, our DRO bounds depend on the ambiguity set geometry and its compatibility with the true loss function. Notably, when using maximum mean discrepancy as a DRO distance metric, our analysis implies, to the best of our knowledge, the first generalization bound in the literature that depends solely on the true loss function, entirely free of any complexity measures or bounds on the hypothesis class.

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The Discrete Moment Problem with Nonconvex Shape Constraints

Cited by 16 publications

References 53 publications

A Unified Framework for Generalized Moment Problems: a Novel Primal-Dual Approach

A Unified Framework for Generalized Moment Problems: a Novel Primal-Dual Approach

On the Impossibility of Statistically Improving Empirical Optimization: A Second-Order Stochastic Dominance Perspective

Generalization Bounds with Minimal Dependency on Hypothesis Class via Distributionally Robust Optimization

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