Accepted and Under Review

Decision-Focused Learning with Directional Gradients. with Michael Huang.

NeurIPS 2025 (to appear)

[Abstract]   [ArXiv]

We propose a novel family of decision-aware surrogate losses, called Perturbation Gradient (PG) losses, for the predict-then-optimize framework. The key idea is to connect the expected downstream decision loss with the directional derivative of a particular plug-in objective, and then approximate this derivative using zeroth order gradient techniques. Unlike the original decision loss which is typically piecewise constant and discontinuous, our new PG losses can be optimized using off-the-shelf gradient-based methods. Most importantly, unlike existing surrogate losses, the approximation error of our PG losses vanishes as the number of samples grows. Hence, optimizing our surrogate loss yields a best-in-class policy asymptotically, even in misspecified settings. This is the first such result in misspecified settings, and we provide numerical evidence confirming our PG losses substantively outperform existing proposals when the underlying model is misspecified.

Beyond Discretization: Learning the Solution Path. with Qiran Dong and Paul Grigas.

Under Review (Oct 2024)

[Abstract]   [ArXiv]

Many applications require minimizing a family of optimization problems indexed by some hyperparameter λ∈Λ to obtain an entire solution path. Traditional approaches proceed by discretizing Λ and solving a series of optimization problems. We propose an alternative approach that parameterizes the solution path with a set of basis functions and solves a \emph{single} stochastic optimization problem to learn the entire solution path. Our method offers substantial complexity improvements over discretization. When using constant-step size SGD, the uniform error of our learned solution path relative to the true path exhibits linear convergence to a constant related to the expressiveness of the basis. When the true solution path lies in the span of the basis, this constant is zero. We also prove stronger results for special cases common in machine learning: When λ∈[−1,1] and the solution path is ν-times differentiable, constant step-size SGD learns a path with ϵ uniform error after at most O(ϵ11−νlog(1/ϵ)) iterations, and when the solution path is analytic, it only requires O(log2(1/ϵ)loglog(1/ϵ)). By comparison, the best-known discretization schemes in these settings require at least O(ϵ−1/2) discretization points (and even more gradient calls). Finally, we propose an adaptive variant of our method that sequentially adds basis functions and demonstrates strong numerical performance through experiments.

Decision-Aware Denoising. with Michael Huang and Paat Rusmevichientong.

Under Review (Oct 2024)

[Abstract]   [SSRN]

Modern decision-making in urban planning, climate change, and healthcare leverages large geospatial and panel datasets. These data are often extremely noisy, resulting in low-quality downstream decisions. We propose a new "light touch" framework to adapt machine learning techniques originally designed for denoising by smoothing to guide decision-making. The cornerstone of our method is a novel estimator of out-of-sample policy performance named the one-shot Variance Gradient Correction (one-shot VGC). By applying the one-shot VGC, we adjust machine learning methods to minimize downstream costs rather than minimizing prediction error or maximizing signal recovery. We uniformly bound the relative error of the one-shot VGC as an estimate of downstream costs by an intuitive measure of solution stability for the problem, plus a term that vanishes as problem dimension increases. Solution stability depends on both the policy and the structure of the downstream optimization problem. We establish bounds for solution stability across three classes of policies and problems: i) regularized plug-in policies for convex feasible regions, ii) unregularized plug-in policies for strongly-convex feasible regions, and iii) unregularized affine plug-in policies for weakly-coupled (potentially non-convex) problems. In all scenarios, we demonstrate that solution stability diminishes (relative to out-of-sample cost) as the dimension of the optimization problem increases. Finally, we present a case study using real traffic accident data from New York City, deploying speed humps to reduce pedestrian injuries. Our "light-touch" decision-aware approach surpasses traditional decision-blind techniques, underscoring that optimal smoothing levels for a denoising algorithm should depend on the downstream decision problem.

Simplifying the Analysis of the Stein Correction_in the Small Data-Large Scale Optimization Regime. with Michael Huang and Paat Rusmevichientong.

Technical Note (Jan 2023)

[Abstract]

This technical note provides an alternate proof of Gupta and Rusmevichientong (2021, Theorem 4.3). The proof presented here is more general, and substantively simpler. It hinges on a new approach to bounding the dependence between different components of the solution of a random linear optimization problem by considering worst-case dual realizations.

Reinforcement Learning for Public Health - Targeted COVID-19 Screening. with Hamsa Bastani and Kimon Drakopoulos.

Invited Chapter to "Artificial Intelligence for Social Impact" (Forthcoming) .

[Abstract]

(This is an invited chapter to a edited book.)

Reinforcement learning is a promising solution to sequential decision-making problems, but its use has largely been limited to simulation environments and e-commerce. This chapter describes a large-scale deployment of reinforcement learning in Greece during the summer of 2020 to adaptively allocate scarce testing resources to incoming passengers amidst the evolving COVID-19 pandemic. Our system, nicknamed Eva, used limited demographic information and recent testing results to guide testing in order to maximize the number of asymptomatic but infected travelers identified over the course of the tourist season. Results from the field evaluation show a marked improvement over other “open- loop” testing strategies and highlight some of the challenges of deploying reinforcement learning in real-world, high-stakes settings.

Debiasing In-Sample Policy Performance for Small-Data, Large-Scale Optimization. with Michael Huang and Paat Rusmevichientong.

Operations Research (Nov. 2022).

[Abstract]   [opre.2022.2377]   [SSRN]   [arXiv:2107.12438]

Motivated by the poor performance of cross-validation in settings where data are scarce, we propose a novel estimator of the out-of-sample performance of a policy in data-driven optimization. Our approach exploits the optimization problem’s sensitivity analysis to estimate the gradient of the optimal objective value with respect to the amount of noise in the data and uses the estimated gradient to debias the policy’s in-sample performance. Importantly, unlike cross-validation techniques, our approach avoids sacrificing data for a test set, utilizes all data when training and, hence, is well-suited to settings where data are scarce. We prove bounds on the bias and variance of our estimator for optimization problems with uncertain objectives but known, potentially non-convex, feasible regions.

For more specialized optimization problems where the feasible region is “weakly-coupled” in a certain sense, we prove stronger results. Specifically, we provide explicit high-probability bounds on the error of our estimator that holds uniformly over a policy class and depends on the problem’s dimension and policy class’s complexity. Importantly, all of our bounds show that under mild conditions, the error of our estimator vanishes as the dimension of the optimization problem grows, even if the amount of available data remains small and constant. Said differently, we prove our estimator performs well in the small-data, large-scale regime.

Finally, we numerically compare our proposed method to state-of-the-art approaches through a case-study on dispatching emergency medical response services using real data. Our method provides more accurate estimates of out-of-sample performance and learns better-performing policies.

Interpretable Operations Research for High-Stakes Decisions: Designing the Greek COVID-19 Testing System. with Hamsa Bastani, Kimon Drakopoulos.

INFORMS Journal on Applied Analytics (Sept. 2022).

[Abstract]   [inte.2022.1128]   [SSRN]

**Winner of the 2021 Daniel H. Wagner Prize for Excellence in the Practice of Advanced Analytics and Operations Research**

In the summer of 2020, in collaboration with the Greek government, we designed and deployed Eva – the first national scale, reinforcement learning system for targeted COVID-19 testing. In this paper, we detail the rationale for three major design/algorithmic elements: Eva’s testing supply chain, estimating COVID-19 prevalence, and test allocation. Specifically, we describe the design of Eva’s supply chain to collect and process thousands of biological samples per day with special emphasis on capacity procurement. Then, we propose a novel, empirical Bayes estimation strategy to estimate COVID-19 prevalence among different passenger types with limited data and showcase how these estimates were instrumental for a variety of downstream decision-making. Finally, we propose a novel, multi-armed bandit algorithm that dynamically allocates tests to arriving passengers in a non-stationary environment with delayed feedback and batched decisions. All of our design and algorithmic choices emphasize the need for transparent reasoning to enable human-in-the-loop analytics. Such transparency was crucial to building trust and buy-in among policymakers and public health experts in a period of global crisis.

Please see also our partner paper – Efficient and Targeted COVID-19 Border Testing via Reinforcement Learning – above for more context on the Eva Project.

Optimization in the Small-Data, Large-Scale Regime.

Invited Chapter to "The Elements of Joint Learning and Optimization in Operations Management" (2022) .

[Abstract]

(This is an invited chapter to a edited book from Springer.)

This chapter introduces the small-data,large-scale optimization regime, an asymptotic setting that arguably better describes certain data-driven optimization applications than the more traditional large-sample regime. We highlight unique phenomena that emerge in the small-data, large-scale regime, and show how these phenomena cause certain traditional data-drivenoptimization algorithms like sample average approximation (SAA) ot fail. We then propose a new debiasing approach that has provably good performance in this regime, highlighting a new path forward for researhch and development into these types of applications.

Efficient and Targeted COVID-19 Border Testing via Reinforcement Learning. with Hamsa Bastani, Kimon Drakopoulos.

Nature (Published online 22 Sept 2021) .

[Abstract]   [10.1038/s41586-021-04014-z]   [SSRN]   [Open Source Algorithm]   [Code Supporting Paper]

**Winner 2021 Pierskalla Best Paper Competition**

**2nd Place 2021 Public Sector Operations Research Best Paper Award**

**Finalist in the Post-Pandemic Supply-Chain and Healthcare Management Conference's Best Paper Competition**

**Spotlight Presentation at the Reinforcement Learning for Real-Life Workshop (ICML 2021)**

(Formerly titled: Deploying a Data-Driven COVID-19 Screening Policy at the Greek Border)

On July 1st, 2020, members of the European Union gradually lifted earlier COVID-19 restrictions on non-essential travel. In response, we designed and deployed “Eva” – a novel reinforcement learning system – across all Greek borders to identify asymptomatic travelers infected with SARS-CoV-2. Eva allocates Greece’s limited testing resources based on demographic characteristics and results from previously tested travelers to (i) limit the influx of new cases and (ii) provide real-time estimates of COVID-19 prevalence to inform border policies. Counterfactual analysis shows that Eva identified 1.85x as many asymptomatic, infected travelers as random surveillance testing, with up to 2-4x as many during peak travel. Moreover, Eva identified approximately 1.25-1.45x as many infected travelers as policies that require similar infrastructure as Eva, but make allocations based on population-level epidemiological metrics (cases/deaths/positivity rates) rather than reinforcement learning. Eva was also able to identify countries with atypically high prevalence earlier than machine learning methods based on epidemiological metrics alone, which allowed Greece to adaptively adjust border policies to prevent additional infected travelers from arriving. Finally, using Eva’s unique cross-country dataset on prevalence in asymptomatic, traveler populations, we show that epidemiological metrics had limited predictive value for the actual prevalence among asymptomatic travelers, and furthermore exhibited strong country-specific idiosyncrasies in summer 2020. Our insights raise serious concerns about internationally proposed border control policies [1] that are country-agnostic and based on population-level epidemiological metrics. Instead, our work paves the way for leveraging reinforcement learning and real-time data for public health goals, such as border control during a pandemic.

Open-source code with a brief overview of the algorithm is also available above.

Please see also:

• Nature News and Views
• Nature Editorial: Greece Used AI to Curb COVID
• USC News
• Webinar with Authors
• Other Popular News Coverage:
  • Entrepreneur
  • EurekAlert!
  • PhocusWire- Travel Weekly Asia
  • USC Marshall News
  • USC Press Release

Data-Pooling in Stochastic Optimization. with Nathan Kallus.

Management Science (Published online 27 March 2021) .

[Abstract]   [10.1287/mnsc.2020.3933]   [SSRN]   [Open-Source Code]

Managing large-scale systems often involves simultaneously solving thousands of unrelated stochastic optimization problems, each with limited data. Intuition suggests one can decouple these unrelated problems and solve them separately without loss of generality. We propose a novel data-pooling algorithm called Shrunken-SAA that disproves this intuition. In particular, we prove that combining data across problems can outperform decoupling, even when there is no a priori structure linking the problems and data are drawn independently. Our approach does not require strong distributional assumptions and applies to constrained, possibly non-convex, non-smooth optimization problems such as vehicle-routing, economic lot-sizing or facility location. We compare and contrast our results to a similar phenomenon in statistics (Stein’s Phenomenon), highlighting unique features that arise in the optimization setting that are not present in estimation. We further prove that as the number of problems grows large, Shrunken-SAA learns if pooling can improve upon decoupling and the optimal amount to pool, even if the average amount of data per problem is fixed and bounded. Importantly, we highlight a simple intuition based on stability that highlights when and why data-pooling offers a benefit, elucidating this perhaps surprising phenomenon. This intuition further suggests that data-pooling offers the most benefits when there are many problems, each of which has a small amount of relevant data. Finally, we demonstrate the practical benefits of data-pooling using real data from a chain of retail drug stores in the context of inventory management.

The Value of Personalized Pricing. with Adam Elmachtoub and Michael Hamilton.

Management Science (Published online 5 April 2021) .

[Abstract]   [10.1287/mnsc.2020.3821]   [SSRN]   [Open-Source Code]

**Finalist in the 2018 INFORMS Service Science Best Paper Award**

**Accepted in The 15th Conference on Web and Internet Economics (WINE), 2019**

Increased availability of high-quality customer information has fueled interest in personalized pricing strategies, i.e., strategies that predict an individual customer’s valuation for a product and then o↵er a price tailored to that customer. While the appeal of personalized pricing is clear, it may also incur large costs in the form of market research, investment in information technology and analytics expertise, and branding risks. In light of these trade-offs, our work studies the value of personalized pricing strategies over a simple single price strategy.

We first provide closed-form lower and upper bounds on the ratio between the profits of an idealized personalized pricing strategy (first-degree price discrimination) and a single price strategy. Our bounds depend on simple statistics of the valuation distribution and shed light on the types of markets for which personalized pricing has little or significant potential value. Second, we consider a feature-based pricing model where customer valuations can be estimated from observed features. We show how to transform our aforementioned bounds into lower and upper bounds on the value of feature-based pricing over single pricing depending on the degree to which the features are informative for the valuation. Finally, we demonstrate how to obtain sharper bounds by incorporating additional information about the valuation distribution (moments or shape constraints) by solving tractable linear optimization problems.

Small-Data, Large-Scale Linear Optimization. with Paat Rusmevichientong.

Management Science (Published online 1 July 2020) .

[Abstract]   [10.1287/mnsc.2019.3554]   [SSRN]   [Open-Source Code]

Optimization applications often depend upon a huge number of uncertain parameters. In many contexts, however, the amount of relevant data per parameter is small, and hence, we may only have imprecise estimates. We term this setting – where the number of uncertainties is large, but all estimates have low precision – the “small-data, large-scale regime.” We formalize a model for this new regime, focusing on optimization problems with uncertain linear objectives. We show that common data-driven methods may perform poorly in this new setting, despite their provably good performance in the traditional large-sample regime. Such methods include sample average approximation, “estimate-then-optimize” policies, data-driven robust optimization, and certain regularized policies.

We then propose a novel framework for selecting a data-driven policy from a given policy class. Like the aforementioned data-driven methods, our policy enjoys provably good performance in the large-sample regime. Unlike, these methods, however, we show that in the small-data, large-scale regime, our data-driven policy performs comparably to an oracle best-in-class policy, provided the policy class and estimates satisfy some mild conditions. We specialize and strengthen this result for linear optimization problems and two natural policy classes: the first inspired by the empirical Bayes literature in statistics and the second by the regularization literature in optimization and machine learning. For both classes, we show that the suboptimality gap between our proposed policy and the oracle policy decays exponentially fast in the number of uncertain parameters, even for a fixed amount of data. Thus, these policies retain the strong large-sample performance of traditional methods, and additionally enjoy provably strong performance in the small-data, large-scale regime. Numerical experiments confirm the significant benefits of our methods.

Maximizing Intervention Effectiveness. with Brian Rongqing Han, Song-Hee Kim, and Hyung Paek.

Management Science (Published online May 2020) .

[Abstract]   [10.1287/mnsc.2019.3537]   [SSRN]

**Finalist in the 2018 Pierskalla Best Paper Award**

**Finalist in the 2018 POMS College of Healthcare and Operations Management (CHOM) Best Paper Competition**

Frequently, policymakers seek to roll out an intervention previously proven effective in a research study, perhaps subject to resource constraints. However, since different subpopulations may respond differently to the same treatment, there is no a priori guarantee that the intervention will be as effective in the targeted population as it was in the study. How then should policymakers target individuals to maximize intervention effectiveness? We propose a novel robust optimization approach that leverages evidence typically available in a published study. Our approach is tractable – real-world instances are easily optimized in minutes with off-the-shelf software – and flexible enough to accommodate a variety of resource and fairness constraints. We compare our approach with current practice by proving performance guarantees for both approaches, which emphasize their structural differences. We also prove an intuitive interpretation of our model in terms of regularization, penalizing differences in the demographic distribution between targeted individuals and the study population. Although the precise penalty depends on the choice of uncertainty set, we show that for special cases we can recover classical penalties from the covariate matching literature on causal inference. Finally, using real data from a large teaching hospital, we compare our approach to common practice in the particular context of reducing emergency department utilization by Medicaid patients through case management. We find that our approach can offer significant benefits over common practice, particularly when the heterogeneity in patient response to the treatment is large.

Popular News Coverage: USC Marshall News

Near-Optimal Bayesian Ambiguity Sets for Distributionally Robust Optimization.

Management Science (Mar. 2019) .

[Abstract]   [10.1287/mnsc.2018.3140]   [Optimization Online]   [Open-Source Code]

We propose a Bayesian framework for assessing the relative strengths of data-driven ambiguity sets in distributionally robust optimization (DRO) when the underlying distribution is defined by a finite-dimensional parameter. The key idea is to measure the relative size between a candidate ambiguity set and a specific, asymptotically optimal set. This asymptotically optimal set is provably the smallest convex ambiguity set that satisfies a particular Bayesian robustness guarantee with respect to a given class of constraints as the amount of data grows large. In other words, it is a subset of any other convex set that satisfies the same guarantee. Using this framework, we prove that existing, popular ambiguity sets based on statistical confidence regions are significantly larger than the asymptotically optimal set with respect to constraints that are concave in the ambiguity– the ratio of their sizes scales with the square root of the dimension of the ambiguity. By contrast, we construct new ambiguity sets that are tractable, satisfy our Bayesian robustness guarantee with finite data and are only a small, constant factor larger than the asymptotically optimal set; we call these sets “Bayesian near-optimal.” We further prove that, asymptotically, solutions to DRO models with our Bayesian near-optimal sets enjoy frequentist robustness properties, despite their smaller size. Finally, our framework yields guidelines for practitioners for selecting between competing ambiguity set proposals in DRO. Computational evidence in portfolio allocation using real and simulated data confirms that our framework, although motivated by asymptotic analysis in a Bayesian setting, provides practical insight into the performance of various DRO models with finite data under frequentist assumptions.

Data-Driven Robust Optimization. with Dimitris Bertsimas and Nathan Kallus.

Mathematical Programming (Feb. 2017) .

[Abstract]   [10.1007/s10107-017-1125-8]   [Open-Source Code]

**Finalist in the 2013 George Nicholson Student Paper Prize**

The last decade has seen an explosion in the availability of data for operations research applications as part of the Big Data revolution. Motivated by this data rich paradigm, we propose a novel schema for utilizing data to design uncertainty sets for robust optimization using statistical hypothesis tests. The approach is flexible and widely applicable, and robust optimization problems built from our new sets are computationally tractable, both theoretically and practically. Furthermore, optimal solutions to these problems enjoy a strong, finite-sample probabilistic guarantee. We also propose concrete guidelines for practitioners and illustrate our approach with applications in portfolio management and queueing. Computational evidence confirms that our data-driven sets significantly outperform conventional robust optimization techniques whenever data is available.

Robust Sample Average Approximation. with Dimitris Bertsimas and Nathan Kallus.

Mathematical Programming (Jun. 2017) .

[Abstract]   [doi:10.1007/s10107-017-1174-z]

**Winner of the 2013 MIT Operations Research Center Best Student Paper Award**

Sample average approximation (SAA) is a widely approach to data-driven decision-making under uncertainty. Under mild assumptions, SAA is both tractable and enjoys strong asymptotic performance guarantees. Similar guarantees, however, do not typically hold in finite samples. In this paper, we propose a modification of SAA, which we term Robust SAA, which retains SAA’s tractability and asymptotic properties and, additionally, enjoys strong finite-sample performance guarantees. The key to our method is linking SAA, distributionally robust optimization, and hypothesis testing of goodness-of-fit. Beyond Robust SAA, this connection provides a unified perspective enabling us to characterize the finite sample and asymptotic guarantees of various other data-driven procedures that are based upon distributionally robust optimization. We present examples from inventory management and portfolio allocation, and demonstrate numerically that our approach outperforms other data-driven approaches in these applications.

A Comparison of Monte Carlo Tree Search and Mathematical Optimization for Large Scale Dynamic Resource Allocation. with Dimitris Bertsimas, John D. Griffith, Mykel Kochenderfer, Velibor Misic, and Robert Moss.

European Journal of Operations Research (Dec. 2017) .

[Abstract]   [doi:10.1016/j.ejor.2017.05.032]

Dynamic resource allocation (DRA) problems are an important class of dynamic stochastic optimization problems that arise in a variety of important real-world applications. DRA problems are notoriously difficult to solve to optimality since they frequently combine stochastic elements with intractably large state and action spaces. Although the artificial intelligence and operations research communities have independently proposed two successful frameworks for solving dynamic stochastic optimization problems—Monte Carlo tree search (MCTS) and mathematical optimization (MO), respectively—the relative merits of these two approaches are not well understood. In this paper, we adapt both MCTS and MO to a problem inspired by tactical wildfire and management and undertake an extensive computational study comparing the two methods on large scale instances in terms of both the state and the action spaces. We show that both methods are able to greatly improve on a baseline, problem-specific heuristic. On smaller instances, the MCTS and MO approaches perform comparably, but the MO approach outperforms MCTS as the size of the problem increases for a fixed computational budget.

Data-Driven Estimation in Equilibrium Using Inverse Optimization. with Dimitris Bertsimas and Ioannis Ch. Paschalidis.

Mathematical Programming (Sep. 2014) .

[Abstract]   [doi:10.1007/s10107-014-0819-4]   [arXiv:1308.3397]   [Open-Source Code]

Equilibrium modeling is common in a variety of fields such as game theory and transportation science. The inputs for these models, however, are often difficult to estimate, while their outputs, i.e., the equilibria they are meant to describe, are often directly observable. By combining ideas from inverse optimization with the theory of variational inequalities, we develop an efficient, data-driven technique for estimating the parameters of these models from observed equilibria. We use this technique to estimate the utility functions of players in a game from their observed actions and to estimate the congestion function on a road network from traffic count data. A distinguishing feature of our approach is that it supports both parametric and nonparametric estimation by leveraging ideas from statistical learning (kernel methods and regularization operators). In computational experiments involving Nash and Wardrop equilibria in a nonparametric setting, we find that a) we effectively estimate the unknown demand or congestion function, respectively, and b) our proposed regularization technique substantially improves the out-of-sample performance of our estimators.

A Course on Advanced Software Tools for Operations Research and Analytics. with Iain Dunning, Angela King, Jerry Kung, Miles Lubin and Jon Silberholz.

INFORMS Transactions on Education (Jan. 2015) .

[Abstract]   [ited.2014.0131]   [Open-Source Code/Materials]

In the age of big data analytics, it is increasingly important for researchers and practitioners to be familiar with methods and software tools for analyzing large data sets, formulating and solving large-scale mathematical optimization models, and sharing solutions using interactive media. Unfortunately, advanced software tools are seldom included in curricula of graduate-level operations research (OR) programs. We describe a course consisting of eight three-hour modules intended to introduce Master’s and PhD students to advanced software tools for OR: Machine Learning in R, Data Wrangling, Visualization, Big Data, Algebraic Modeling with JuMP, High-Performance and Distributed Computing, Internet and Databases, and Advanced Mixed Integer Linear Programming (MILP) Techniques. For each module, we outline content, provide course materials, summarize student feedback, and share lessons learned from two iterations of the course. Student feedback was very positive, and all students reported that the course equipped them with software skills useful for their own research. We believe our course materials could serve as a template for the development of effective OR software tools courses and discuss how they could be adapted to other educational settings.

Inverse Optimization: A New Perspective on the Black-Litterman Model. with Dimitris Bertsimas and Ioannis Ch. Paschalidis.

Operations Research (Nov. 2012) .

[Abstract]   [doi:10.1287/opre.1120.1115]

The Black-Litterman (BL) model is a widely used asset allocation model in the financial industry. In this paper, we provide a new perspective. The key insight is to replace the statistical framework in the original approach with ideas from inverse optimization. This insight allows us to significantly expand the scope and applicability of the BL model. We provide a richer formulation that, unlike the original model, is flexible enough to incorporate investor information on volatility and market dynamics. Equally importantly, our approach allows us to move beyond the traditional mean-variance paradigm of the original model and construct “BL”-type estimators for more general notions of risk such as coherent risk measures. Computationally, we introduce and study two new “BL”-type estimators and their corresponding portfolios: a mean variance inverse optimization (MV-IO) portfolio and a robust mean variance inverse optimization (RMV-IO) portfolio. These two approaches are motivated by ideas from arbitrage pricing theory and volatility uncertainty. Using numerical simulation and historical backtesting, we show that both methods often demonstrate a better risk-reward trade-off than their BL counterparts and are more robust to incorrect investor views.