Introduction to stochastic programming

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High-dimensional Statistics: A Non-asymptotic Viewpoint, Martin J. Wainwright, Cambridge University Press, 2019, xvii 552 pages, £57.99, hardback ISBN: 978-1-1084-9802-9

The concepts of risk-aversion, chance-constrained optimization, and robust optimization have developed significantly over the last decade. Statistical learning community has also witnessed a rapid theoretical and applied growth by relying on these concepts. A modeling framework, called distributionally robust optimization (DRO), has recently received significant attention in both the operations research and statistical learning communities. This paper surveys main concepts and contributions to DRO, and its relationships with robust optimization, risk-aversion, chance-constrained optimization, and function regularization.

Distributionally Robust Optimization: A Review

Optimization under uncertainty in the era of big data and deep learning: When machine learning meets mathematical programming

Many decision problems in science, engineering and economics are affected by uncertain parameters whose distribution is only indirectly observable through samples. The goal of data-driven decision-making is to learn a decision from finitely many training samples that will perform well on unseen test samples. This learning task is difficult even if all training and test samples are drawn from the same distribution---especially if the dimension of the uncertainty is large relative to the training sample size. Wasserstein distributionally robust optimization seeks data-driven decisions that perform well under the most adverse distribution within a certain Wasserstein distance from a nominal distribution constructed from the training samples. In this tutorial we will argue that this approach has many conceptual and computational benefits. Most prominently, the optimal decisions can often be computed by solving tractable convex optimization problems, and they enjoy rigorous out-of-sample and asymptotic consistency guarantees. We will also show that Wasserstein distributionally robust optimization has interesting ramifications for statistical learning and motivates new approaches for fundamental learning tasks such as classification, regression, maximum likelihood estimation or minimum mean square error estimation, among others.

Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning

We present a new distributionally robust optimization model called robust stochastic optimization (RSO), which unifies both scenario-tree-based stochastic linear optimization and distributionally r...

Robust Stochastic Optimization Made Easy with RSOME

We provide an exact deterministic reformulation for data-driven chance constrained programs over Wasserstein balls. For individual chance constraints as well as joint chance constraints with right-hand side uncertainty, our reformulation amounts to a mixed-integer conic program. In the special case of a Wasserstein ball with the $1$-norm or the $\infty$-norm, the cone is the nonnegative orthant, and the chance constrained program can be reformulated as a mixed-integer linear program. Using our reformulation, we show that two popular approximation schemes based on the conditional-value-at-risk and the Bonferroni inequality can perform poorly in practice and that these two schemes are generally incomparable with each other.

Data-Driven Chance Constrained Programs over Wasserstein Balls

Distributionally robust optimization has grown into one of the most popular approaches in addressing optimization under uncertainty. As its key ingredient, the ambiguity set specifies several types...

Distributionally Robust Optimization with Infinitely Constrained Ambiguity Sets

We study the adaptive distributionally robust hub location problem with multiple commodities under demand and cost uncertainty in both uncapacitated and capacitated cases. The hub location decision...

Distributionally Robust Hub Location

The distributionally robust Markov Decision Process (MDP) approach asks for a distributionally robust strategy that achieves the maximal expected total reward under the most adversarial distributio...

Zhi Chen

Papers

Robust Stochastic Optimization Made Easy with RSOME

Data-Driven Chance Constrained Programs over Wasserstein Balls

Distributionally Robust Optimization with Infinitely Constrained Ambiguity Sets

Distributionally Robust Hub Location

Distributionally Robust Optimization for Sequential Decision-Making