Robust and Heterogenous Odds Ratio: Estimating Price Sensitivity for Unbought Items

Problem definition: Mining for heterogeneous responses to an intervention is a crucial step for data-driven operations, for instance to personalize treatment or pricing. We investigate how to estimate price sensitivity from transaction-level data. In causal inference terms, we estimate heterogeneous treatment effects when (a) the response to treatment (here, whether a customer buys a product) is binary, and (b) treatment assignments are partially observed (here, full information is only available for purchased items). Methodology/Results: We propose a recursive partitioning procedure to estimate heterogeneous odds ratio, a widely used measure of treatment effect in medicine and social sciences. We integrate an adversarial imputation step to allow for robust inference even in presence of partially observed treatment assignments. We validate our methodology on synthetic data and apply it to three case studies from political science, medicine, and revenue management. Managerial Implications: Our robust heterogeneous odds ratio estimation method is a simple and intuitive tool to quantify heterogeneity in patients or customers and personalize interventions, while lifting a central limitation in many revenue management data

Sparse PCA With Multiple Components

Sparse Principal Component Analysis (sPCA) is a cardinal technique for obtaining combinations of features, or principal components (PCs), that explain the variance of high-dimensional datasets in an interpretable manner. This involves solving a sparsity and orthogonality constrained convex maximization problem, which is extremely computationally challenging. Most existing works address sparse PCA via methods-such as iteratively computing one sparse PC and deflating the covariance matrix-that do not guarantee the orthogonality, let alone the optimality, of the resulting solution when we seek multiple mutually orthogonal PCs. We challenge this status by reformulating the orthogonality conditions as rank constraints and optimizing over the sparsity and rank constraints simultaneously. We design tight semidefinite relaxations to supply high-quality upper bounds, which we strengthen via additional second-order cone inequalities when each PC's individual sparsity is specified. Further, we derive a combinatorial upper bound on the maximum amount of variance explained as a function of the support. We exploit these relaxations and bounds to propose exact methods and rounding mechanisms that, together, obtain solutions with a bound gap on the order of 0%-15% for real-world datasets with p = 100s or 1000s of features and r \in {2, 3} components. Numerically, our algorithms match (and sometimes surpass) the best performing methods in terms of fraction of variance explained and systematically return PCs that are sparse and orthogonal. In contrast, we find that existing methods like deflation return solutions that violate the orthogonality constraints, even when the data is generated according to sparse orthogonal PCs. Altogether, our approach solves sparse PCA problems with multiple components to certifiable (near) optimality in a practically tractable fashion.Comment: Updated version with improved algorithmics and a new section containing a generalization of the Gershgorin circle theorem; comments or suggestions welcom

Prediction with Missing Data

Missing information is inevitable in real-world data sets. While imputation is well-suited and theoretically sound for statistical inference, its relevance and practical implementation for out-of-sample prediction remains unsettled. We provide a theoretical analysis of widely used data imputation methods and highlight their key deficiencies in making accurate predictions. Alternatively, we propose adaptive linear regression, a new class of models that can be directly trained and evaluated on partially observed data, adapting to the set of available features. In particular, we show that certain adaptive regression models are equivalent to impute-then-regress methods where the imputation and the regression models are learned simultaneously instead of sequentially. We validate our theoretical findings and adaptive regression approach with numerical results with real-world data sets

A Stochastic Benders Decomposition Scheme for Large-Scale Data-Driven Network Design

Network design problems involve constructing edges in a transportation or supply chain network to minimize construction and daily operational costs. We study a data-driven version of network design where operational costs are uncertain and estimated using historical data. This problem is notoriously computationally challenging, and instances with as few as fifty nodes cannot be solved to optimality by current decomposition techniques. Accordingly, we propose a stochastic variant of Benders decomposition that mitigates the high computational cost of generating each cut by sampling a subset of the data at each iteration and nonetheless generates deterministically valid cuts (as opposed to the probabilistically valid cuts frequently proposed in the stochastic optimization literature) via a dual averaging technique. We implement both single-cut and multi-cut variants of this Benders decomposition algorithm, as well as a k-cut variant that uses clustering of the historical scenarios. On instances with 100-200 nodes, our algorithm achieves 4-5% optimality gaps, compared with 13-16% for deterministic Benders schemes, and scales to instances with 700 nodes and 50 commodities within hours. Beyond network design, our strategy could be adapted to generic two-stage stochastic mixed-integer optimization problems where second-stage costs are estimated via a sample average

Algorithmic advancements in discrete optimization : applications to machine learning and healthcare operations

Thesis: Ph. D., Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, May, 2020Cataloged from the official PDF of thesis.Includes bibliographical references (pages 235-253).In the next ten years, hospitals will operate like air-traffic control centers whose role is to coordinate care across multiple facilities. Consequently, the future of hospital operations will have three salient characteristics. First, data. The ability to process, analyze and exploit data effectively will become a vital skill for practitioners. Second, a holistic approach, since orchestrating care requires the concurrent optimization of multiple resources, services, and time scales. Third, real-time personalized decisions, to respond to the increasingly closer monitoring of patients. To support this transition and transform our healthcare system towards better outcomes at lower costs, research in operations and analytics should address two concurrent goals: First, develop new methods and algorithms for decision-making in a data-rich environment, which answer key concerns from practitioners and regulators, such as reliability, interpretability, and fairness.Second, put its models and algorithms to the test of practice, to ensure a path towards implementation and impact. Accordingly, this thesis is comprised of two parts. The first three chapters present methodological contributions to the discrete optimization literature, with particular emphasis on problems emerging from machine learning under sparsity. Indeed, the most important operational decision-making problems are by nature discrete and their sizes have increased with the widespread adoption of connected devices and sensors. In particular, in machine learning, the gigantic amount of data now available contrasts with our limited cognitive abilities. Hence, sparse models, i.e., which only involve a small number of variables, are needed to ensure human understanding. The last two chapters present applications and implementation of machine learning and discrete optimization methods to improve operations at a major academic hospital.From raw electronic health records of patients, we build predictive models to predict patient flows and prescriptive models to optimize patient-bed assignment in real-time. More importantly, we implement our models in a 600-bed institution. Our impact is two-fold: methodological and operational. Integrating advanced analytics in their daily operations and building a data-first culture constitutes a major paradigm shift.by Jean Pauphilet.Ph. D.Ph.D. Massachusetts Institute of Technology, Sloan School of Management, Operations Research Cente

Robust combination testing: methods and application to COVID-19 detection

Simple and affordable testing tools are often not accurate enough to be operationally relevant. For COVID-19 detection, rapid point-of-care tests are cheap and provide results in minutes, but largely fail policymakers’ accuracy requirements. We propose an analytical methodology, based on robust optimization, that identifies optimal combinations of results from cheap tests for increased predictive accuracy. This methodological tool allows policymakers to credibly quantify the benefits from combination testing and thus break the trade-off between cost and accuracy. Our methodology is robust to noisy and partially missing input data and incorporates operational constraints—relevant considerations in practice. We apply our methodology to two datasets containing individual-level results of multiple COVID-19 rapid antibody and antigen tests, respectively, to generate Pareto-dominating receiver operating characteristic (ROC) curves. We find that combining only three rapid tests increases out-of-sample area under the curve (AUC) by 4% (6%) compared with the best performing individual test for antibody (antigen) detection. We also find that a policymaker who requires a specificity of at least 0.95 can improve sensitivity by 8% and 2% for antibody and antigen testing, respectively, relative to available combination testing heuristics. Our numerical analysis demonstrates that robust optimization is a powerful tool to avoid overfitting, accommodate missing data, and improve out-of-sample performance. Based on our analytical and empirical results, policymakers should consider approving and deploying a curated combination of cheap point-of-care tests in settings where ‘gold standard’ tests are too expensive or too slow

Robust convex optimization: A new perspective that unifies and extends

Abstract Robust convex constraints are difficult to handle, since finding the worst-case scenario is equivalent to maximizing a convex function. In this paper, we propose a new approach to deal with such constraints that unifies most approaches known in the literature and extends them in a significant way. The extension is either obtaining better solutions than the ones proposed in the literature, or obtaining solutions for classes of problems unaddressed by previous approaches. Our solution is based on an extension of the Reformulation-Linearization-Technique, and can be applied to general convex inequalities and general convex uncertainty sets. It generates a sequence of conservative approximations which can be used to obtain both upper- and lower- bounds for the optimal objective value. We illustrate the numerical benefit of our approach on a robust control and robust geometric optimization example

From predictions to prescriptions: A data-driven response to COVID-19

Abstract The COVID-19 pandemic has created unprecedented challenges worldwide. Strained healthcare providers make difficult decisions on patient triage, treatment and care management on a daily basis. Policy makers have imposed social distancing measures to slow the disease, at a steep economic price. We design analytical tools to support these decisions and combat the pandemic. Specifically, we propose a comprehensive data-driven approach to understand the clinical characteristics of COVID-19, predict its mortality, forecast its evolution, and ultimately alleviate its impact. By leveraging cohort-level clinical data, patient-level hospital data, and census-level epidemiological data, we develop an integrated four-step approach, combining descriptive, predictive and prescriptive analytics. First, we aggregate hundreds of clinical studies into the most comprehensive database on COVID-19 to paint a new macroscopic picture of the disease. Second, we build personalized calculators to predict the risk of infection and mortality as a function of demographics, symptoms, comorbidities, and lab values. Third, we develop a novel epidemiological model to project the pandemic’s spread and inform social distancing policies. Fourth, we propose an optimization model to re-allocate ventilators and alleviate shortages. Our results have been used at the clinical level by several hospitals to triage patients, guide care management, plan ICU capacity, and re-distribute ventilators. At the policy level, they are currently supporting safe back-to-work policies at a major institution and vaccine trial location planning at Janssen Pharmaceuticals, and have been integrated into the US Center for Disease Control’s pandemic forecast