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Problem definition: Data analytics models and machine learning algorithms are increasingly deployed to support consequential decision-making processes, from deciding which applicants will receive job offers and loans to university enrollments and medical interventions. However, recent studies show these models may unintentionally amplify human bias and yield significant unfavorable decisions to specific groups. Methodology/results: We propose a distributionally robust classification model with a fairness constraint that encourages the classifier to be fair in the equality of opportunity criterion. We use a type- ∞ Wasserstein ambiguity set centered at the empirical distribution to represent distributional uncertainty and derive a conservative reformulation for the worst-case equal opportunity unfairness measure. We show that the model is equivalent to a mixed binary conic optimization problem, which standard off-the-shelf solvers can solve. We propose a convex, hinge-loss-based model for large problem instances whose reformulation does not incur binary variables to improve scalability. Moreover, we also consider the distributionally robust learning problem with a generic ground transportation cost to hedge against the label and sensitive attribute uncertainties. We numerically examine the performance of our proposed models on five real-world data sets related to individual analysis. Compared with the state-of-the-art methods, our proposed approaches significantly improve fairness with negligible loss of predictive accuracy in the testing data set. Managerial implications: Our paper raises awareness that bias may arise when predictive models are used in service and operations. It generally comes from human bias, for example, imbalanced data collection or low sample sizes, and is further amplified by algorithms. Incorporating fairness constraints and the distributionally robust optimization (DRO) scheme is a powerful way to alleviate algorithmic biases. Funding: This work was supported by the National Science Foundation [Grants 2342505 and 2343869] and the Chinese University of Hong Kong [Grant 4055191]. Supplemental Material: The online appendices are available at https://doi.org/10.1287/msom.2022.0230. [ABSTRACT FROM AUTHOR]

We study two-stage stochastic optimization problems with random recourse, where the coefficients of the adaptive decisions involve uncertain parameters. To deal with the infinite-dimensional recourse decisions, we propose a scalable approximation scheme via piecewise linear and piecewise quadratic decision rules. We develop a data-driven distributionally robust framework with two layers of robustness to address distributional uncertainty. We also establish out-of-sample performance guarantees for the proposed scheme. Applying known ideas, the resulting optimization problem can be reformulated as an exact copositive program that admits semidefinite programming approximations. We design an iterative decomposition algorithm, which converges under some regularity conditions, to reduce the runtime needed to solve this program. Through numerical examples for various known operations management applications, we demonstrate that our method produces significantly better solutions than the traditional sample-average approximation scheme especially when the data are limited. For the problem instances for which only the recourse cost coefficients are random, our method exhibits slightly inferior out-of-sample performance but shorter runtimes compared with a competing approach. History: Accepted by Nicola Secomandi, Area Editor for Stochastic Models & Reinforcement Learning. Funding: This work was supported by the National Science Foundation [Grants 2342505 and 2343869]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information (https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2021.0306) as well as from the IJOC GitHub software repository (https://github.com/INFORMSJoC/2021.0306). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/. [ABSTRACT FROM AUTHOR]

Traditional clustering algorithms such as k-means and vanilla spectral clustering are known to deteriorate significantly in the presence of outliers. Several previous works in literature have proposed robust variants of these algorithms; however, they do not provide any theoretical guarantees. Extending previous clustering literature on Gaussian mixture models, in their paper "A Robust Spectral Clustering Algorithm for Sub-Gaussian Mixture Models with Outliers," Prateek R. Srivastava, Purnamrita Sarkar, and Grani A. Hanasusanto developed a new spectral clustering algorithm and provided error bounds for the algorithm under a general sub-Gaussian mixture model setting with outliers. Surprisingly, their derived error bound matches with the best-known bound for semidefinite programs under the same setting without outliers. Numerical experiments on a variety of simulated and real-world data sets further demonstrate that their algorithm is less sensitive to outliers compared with other state-of-the-art algorithms. We consider the problem of clustering data sets in the presence of arbitrary outliers. Traditional clustering algorithms such as k-means and spectral clustering are known to perform poorly for data sets contaminated with even a small number of outliers. In this paper, we develop a provably robust spectral clustering algorithm that applies a simple rounding scheme to denoise a Gaussian kernel matrix built from the data points and uses vanilla spectral clustering to recover the cluster labels of data points. We analyze the performance of our algorithm under the assumption that the "good" data points are generated from a mixture of sub-Gaussians (we term these "inliers"), whereas the outlier points can come from any arbitrary probability distribution. For this general class of models, we show that the misclassification error decays at an exponential rate in the signal-to-noise ratio, provided the number of outliers is a small fraction of the inlier points. Surprisingly, this derived error bound matches with the best-known bound for semidefinite programs (SDPs) under the same setting without outliers. We conduct extensive experiments on a variety of simulated and real-world data sets to demonstrate that our algorithm is less sensitive to outliers compared with other state-of-the-art algorithms proposed in the literature. Funding: G. A. Hanasusanto was supported by the National Science Foundation Grants NSF ECCS-1752125 and NSF CCF-2153606. P. Sarkar gratefully acknowledges support from the National Science Foundation Grants NSF DMS-1713082, NSF HDR-1934932 and NSF 2019844. Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2022.2317. [ABSTRACT FROM AUTHOR]

We study the problem of finding the Löwner–John ellipsoid (i.e., an ellipsoid with minimum volume that contains a given convex set). We reformulate the problem as a generalized copositive program and use that reformulation to derive tractable semidefinite programming approximations for instances where the set is defined by affine and quadratic inequalities. We prove that, when the underlying set is a polytope, our method never provides an ellipsoid of higher volume than the one obtained by scaling the maximum volume-inscribed ellipsoid. We empirically demonstrate that our proposed method generates high-quality solutions and can be solved much faster than solving the problem to optimality. Furthermore, we outperform the existing approximation schemes in terms of solution time and quality. We present applications of our method to obtain piecewise linear decision rule approximations for dynamic distributionally robust problems with random recourse and to generate ellipsoidal approximations for the set of reachable states in a linear dynamical system when the set of allowed controls is a polytope. [ABSTRACT FROM AUTHOR]

We study robust convex quadratic programs where the uncertain problem parameters can contain both continuous and integer components. Under the natural boundedness assumption on the uncertainty set, we show that the generic problems are amenable to exact copositive programming reformulations of polynomial size. These convex optimization problems are NP-hard but admit a conservative semidefinite programming (SDP) approximation that can be solved efficiently. We prove that the popular approximate S-lemma method—which is valid only in the case of continuous uncertainty—is weaker than our approximation. We also show that all results can be extended to the two-stage robust quadratic optimization setting if the problem has complete recourse. We assess the effectiveness of our proposed SDP reformulations and demonstrate their superiority over the state-of-the-art solution schemes on instances of least squares, project management, and multi-item newsvendor problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of operating a battery storage unit in a home with a rooftop solar photovoltaic (PV) system so as to minimize expected long-run electricity costs under uncertain electricity usage, PV generation, and electricity prices. Solving this dynamic program using standard techniques is computationally burdensome, and is often complicated by the difficulty of estimating conditional distributions from sparse data. To overcome these challenges, we implement a data-driven dynamic programming (DDP) algorithm that uses historical data observations to generate empirical conditional distributions and approximate the cost-to-go function. Then, we formulate two robust data-driven dynamic programming (RDDP) algorithms that consider the worst-case expected cost over a set of conditional distributions centered at the empirical distribution, and within a given Chi-square or Wasserstein distance, respectively. We test our algorithms using data from homes with rooftop PV in Austin, Texas. Numerical results reveal that DDP and RDDP outperform common existing methods with acceptable computational effort. Finally, we show that implementation of these superior operational algorithms significantly raises the break-even battery cost under which a homeowner is incentivized to invest in a residential battery rather than participate in a feed-in tariff or net energy metering program. [ABSTRACT FROM AUTHOR]

In this paper, we show that the popular K-means clustering problem can equivalently be reformulated as a conic program of polynomial size. The arising convex optimization problem is NP-hard, but amenable to a tractable semidefinite programming (SDP) relaxation that is tighter than the current SDP relaxation schemes in the literature. In contrast to the existing schemes, our proposed SDP formulation gives rise to solutions that can be leveraged to identify the clusters. We devise a new approximation algorithm for K-means clustering that utilizes the improved formulation and empirically illustrate its superiority over the state-of-the-art solution schemes. [ABSTRACT FROM AUTHOR]

In data-driven inverse optimization an observer aims to learn the preferences of an agent who solves a parametric optimization problem depending on an exogenous signal. Thus, the observer seeks the agent's objective function that best explains a historical sequence of signals and corresponding optimal actions. We focus here on situations where the observer has imperfect information, that is, where the agent's true objective function is not contained in the search space of candidate objectives, where the agent suffers from bounded rationality or implementation errors, or where the observed signal-response pairs are corrupted by measurement noise. We formalize this inverse optimization problem as a distributionally robust program minimizing the worst-case risk that the predicted decision (i.e., the decision implied by a particular candidate objective) differs from the agent's actual response to a random signal. We show that our framework offers rigorous out-of-sample guarantees for different loss functions used to measure prediction errors and that the emerging inverse optimization problems can be exactly reformulated as (or safely approximated by) tractable convex programs when a new suboptimality loss function is used. We show through extensive numerical tests that the proposed distributionally robust approach to inverse optimization attains often better out-of-sample performance than the state-of-the-art approaches. [ABSTRACT FROM AUTHOR]