Distributionally Robust Optimization Research Articles

Wasserstein distributionally robust optimization (DRO) has gained prominence in operations research and machine learning as a powerful method for achieving solutions with favorable out-of-sample performance. Two compelling explanations for its success are the generalization bounds derived from Wasserstein DRO and its equivalence to regularization schemes commonly used in machine learning. However, existing results on generalization bounds and regularization equivalence are largely limited to settings where the Wasserstein ball is of a specific type, and the decision criterion takes certain forms of expected functions. In this paper, we show that generalization bounds and regularization equivalence can be obtained in a significantly broader setting, where the Wasserstein ball is of a general type and the decision criterion accommodates any form, including general risk measures. This not only addresses important machine learning and operations management applications but also expands to general decision-theoretical frameworks previously unaddressed by Wasserstein DRO. Our results are strong in that the generalization bounds do not suffer from the curse of dimensionality and the equivalency to regularization is exact. As a by-product, we show that Wasserstein DRO coincides with the recent max-sliced Wasserstein DRO for any decision criterion under affine decision rules, resulting in both being efficiently solvable as convex programs via our general regularization results. These general assurances provide a strong foundation for expanding the application of Wasserstein DRO across diverse domains of data-driven decision problems. This paper was accepted by Chung Piaw Teo, optimization. Funding: This work was supported by the National Natural Science Foundation of China [Grants 12371476 and 71921001] and the Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2023-05829]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/mnsc.2023.03895 .

In this study, we address the problem of allocating Intensive Care Unit (ICU) beds among hospitals in a hospital system during a pandemic. The goal is to minimize the total costs associated with deaths, patient transfers, and bed setup by dynamically transferring patients among hospitals and admitting and discharging them from ICUs. The problem is initially formulated as a stochastic dynamic programming problem using Markov chains to model the evolution of patient health conditions. We propose a rolling horizon approach with estimation updating, integrating Bayesian methods to dynamically adjust ICU capacity and patient allocation decisions based on real-time data, thus enhancing responsiveness and accuracy in managing uncertainties in patient arrivals and optimizing resource allocation under varying healthcare demands. However, the traditional problem assumes exogenous uncertainty, whereas in reality, the availability of hospitals can influence patients’ decisions to seek medical services, and therefore, affect the number of patients arriving at hospitals. To address this issue, we propose a distributionally robust optimization (DRO) method that considers the dependence of the ambiguity set on decisions. We reformulate the decision-dependent DRO model as a mixed-integer programming problem. Using a real-world case study of a pandemic, we compare the DRO method with two benchmarks and show that the method is computationally scalable and results in a lower number of deaths.

Single-level reformulations of (nonconvex) distributionally robust optimization (DRO) problems are often intractable, as they contain semi-infinite dual constraints. Based on such a semi-infinite reformulation, we present a safe approximation that allows for the computation of feasible solutions for DROs that depend on nonconvex multivariate simple functions. Moreover, the approximation allows to address ambiguity sets that can incorporate information on moments as well as confidence sets. The typical strong assumptions on the structure of the underlying constraints, such as convexity in the decisions or concavity in the uncertainty found in the literature were, at least in part, recently overcome in [16]. We start from the duality-based reformulation approach in [16] that can be applied for DRO constraints based on simple functions that are univariate in the uncertainty parameters. We significantly extend their approach to multivariate simple functions, which leads to a considerably wider applicability of the proposed reformulation approach. In order to achieve algorithmic tractability, the presented safe approximation is then realized by a discretized counterpart for the semi-infinite dual constraints. The approximation leads to a computationally tractable mixed-integer positive semidefinite problem for which state-of-the-art software implementations are readily available. The tractable safe approximation provides sufficient conditions for distributional robustness of the original problem, i.e., obtained solutions are provably robust.

In this paper, we establish that inequality systems involving a class of high-order separable polynomials exhibit hidden convexity: Their associated epigraphical sets are convex sets even when the underlying polynomials are not convex. This feature of hidden convexity, combined with polynomial separability, enables the derivation of a numerically tractable sum-of-squares (SOS) representation for certifying polynomial non-negativity over a set described by homogeneous separable quadratic inequalities. We apply this representation result to a class of distributionally robust optimization (DRO) problems, where the ambiguity sets are defined via high-order separable polynomial inequalities. We show that these DRO problems admit dual formulations with hidden convex constraint systems, making the SOS-based reformulation not only possible but exact, as it produces a single semi-definite program with the same optimal values. We validate the numerical tractability and applicability of our reformulation results through numerical experiments in portfolio optimization with real market data.

The restoration of modern power systems after large-scale outages poses significant challenges due to the increasing integration of renewable energy sources (RES) and electric vehicles (EVs), both of which introduce new dimensions of uncertainty and flexibility. This paper presents a Hierarchical Modern Power System Restoration (HMPSR) model that employs a two-level architecture to enhance restoration efficiency and system resilience. At the upper level, Graph Neural Networks (GNNs) are used to predict fault locations and optimize network topology by analyzing the spatial and topological features of the grid. At the lower level, Distributionally Robust Optimization (DRO) is applied to manage uncertainty in generation and demand through scenario-based dispatch planning. The model specifically considers solar and wind power as the primary RES, and incorporates both grid-connected and mobile EVs as flexible energy resources to support the restoration process. Simulation results on an enhanced IEEE 33-bus test system demonstrate that the HMPSR model reduces restoration time by 18.6% and total cost by 15.4%, while maintaining a Grid Stability Index above 85% under high variability conditions. These results confirm the effectiveness of a tightly integrated, hierarchical strategy for power system restoration, providing a robust and adaptive framework for real-world deployment.

In this paper, we propose a sparse distributionally robust optimization (DRO) model incorporating the Conditional Value-at-Risk (CVaR) measure to control tail risks in uncertain environments. The model utilizes sparsity to reduce transaction costs and enhance operational efficiency. We reformulate the problem as a Min-Max-Min optimization and convert it into an equivalent non-smooth minimization problem. To address this computational challenge, we develop an approximate discretization (AD) scheme for the underlying continuous random vector and prove its convergence to the original non-smooth formulation under mild conditions. The resulting problem can be efficiently solved using a subgradient method. While our analysis focuses on CVaR penalty, this approach is applicable to a broader class of non-smooth convex regularizers. The experimental results on the portfolio selection problem confirm the effectiveness and scalability of the proposed AD algorithm.

Initially developed in 1937, Operational Research focused on constructing deterministic models in order to describe and analyze real-world problems to aid decision-making. It was not until the early 1950s when uncertainty was incorporated into these models. Traditionally, two primary disciplines have been employed: Stochastic Optimization (SO) and Robust Optimization (RO). In SO, uncertain parameters are modeled as random variables with known probability distributions. However, there has been significant criticism of its optimistic results in recent years. In contrast, RO addresses uncertainty by considering the worst-case scenarios, leading to over-conservative decisions for other more likely scenarios. An emerging paradigm Distributionally Robust Optimization (DRO) has recently garnered significant attention due to its potential to address the limitations of both SO and RO. DRO serves as a unifying framework by introducing the concept of an ambiguity set, which encompasses a family of distributions deemed close-enough to a reference distribution.

This study presents a process-centric hybrid energy management framework tailored for large-scale smart mining operations. The framework addresses three major challenges: (i) multi-source uncertainty propagation, (ii) cross-process energy coupling, and (iii) time-varying, safety-critical operational constraints. The energy scheduling problem is formulated as a process-constrained, multi-period optimization under uncertainty, explicitly modeling the spatio-temporal correlations among renewable power generation, ventilation loads, dewatering demands, and blasting energy requirements. To tackle high-dimensional uncertainties with non-Gaussian distributions, a Wasserstein metric-based distributionally robust optimization (DRO) model is constructed. The ambiguity set is dynamically refined through adaptive scenario generation and clustering, capturing worst-case energy supply-demand mismatches. The objective function jointly minimizes total energy cost, carbon emissions, and process-specific operational risks, subject to nonlinear thermodynamic process constraints, piecewise convex ventilation characteristics, and interdependent hydraulic-ventilation-thermal (HVT) processes. Mining safety regulations are integrated via chance constraints, embedding safety-critical margins related to pressure, airflow, and gas concentration. To alleviate the computational burden caused by nested risk formulations, a Primal-Dual Reformulated Distributionally Robust Process Scheduling (PDR-DRPS) algorithm is proposed. This method recursively updates process-coupled dual variables, enabling fast convergence within joint physical-energy feasible subspaces. The proposed framework is validated using a synthetic open-pit mining benchmark incorporating real-world meteorological data, empirical process dynamics, and regulatory thresholds. Numerical results indicate a 25.4% reduction in operational costs, a 31.2% cut in carbon emissions, and consistent adherence to safety constraints within a 3% tolerance under all uncertainty scenarios. Sensitivity analysis further highlights that process inertia and time delays significantly amplify uncertainty propagation, underscoring the necessity of process-aware robust energy scheduling in safety-critical industrial systems. The framework offers a generalizable paradigm applicable to smart mining, tunnel construction, and underground industrial infrastructures.

In the context of building a new type of power system, the optimal operation of high-proportion new-energy distribution networks (HNEDNs) is a current hot topic. In this paper, a stochastic distribution robust optimization method for HNEDNs that considers energy-storage refinement modeling is proposed. First, an energy-storage lifetime loss model based on the rainfall-counting method is constructed, and then an optimal operation model of an HNEDN considering energy storage refinement modeling is constructed, aiming to minimize the total operation cost while taking into account the energy cost and the penalty cost of abandoning wind and solar power. Then, a source-load uncertainty model of HNEDN is constructed based on the Wasserstein distance and conditional value at risk (CvaR) theory, and the HNEDN optimization model is reconstructed based on the stochastic distribution robust optimization method; based on this, the multiple linearization technique is introduced to approximate the reconstructed model, which aims to both reduce the difficulty in solving the model and ensure the quality of the solution. Finally, the modified IEEE 33-bus power distribution system is used as an example for case analysis, and the simulation results show that the method presented in this paper, through reducing the loss of life in the battery storage device, can reduce the average daily energy storage depreciation cost compared to an HNEDN optimization method that does not take the energy storage life loss into account; this, in turn, reduces the total operating cost of the system. In addition, the stochastic distribution robust optimization method used in this paper can adaptively adjust the economy and robustness of the HNEDN operation strategy according to the confidence level and the available historical sample data on new energy-output prediction errors to obtain the optimal HNEDN operation strategy when compared with other uncertainty treatment methods.

Blood product shortages and wastage costs remain persistent challenges for hospital blood banks (HBBs) due to uncertain demand, perishability, and limited supply. Existing approaches often fail to offer robust performance under distributional shifts or partial data availability and tend to yield overly conservative solutions. This research develops a stochastic optimization (SO) framework for blood product ordering and allocation decisions under uncertain urgent and elective demands. Two robust counterparts are proposed: (i) a hybrid robust optimization (HRO) model combining conditional value at risk (CVaR) and P-Robustness, and (ii) a distributionally robust optimization (DRO) model. Their performance is evaluated relative to each other and a conventional SO model. Numerical experiments show that the DRO model achieves the greatest cost savings, with total costs 5% lower than the SO model and 2% lower than the HRO model. The results provide managerial insights and demonstrate the effectiveness of distributional robustness in managing blood inventory under uncertainty.

Distributionally robust optimization (DRO) studies decision problems under uncertainty where the probability distribution governing the uncertain problem parameters is itself uncertain. A key component of any DRO model is its ambiguity set, that is, a family of probability distributions consistent with any available structural or statistical information. DRO seeks decisions that perform best under the worst distribution in the ambiguity set. This worst case criterion is supported by findings in psychology and neuroscience, which indicate that many decision-makers have a low tolerance for distributional ambiguity. DRO is rooted in statistics, operations research and control theory, and recent research has uncovered its deep connections to regularization techniques and adversarial training in machine learning. This survey presents the key findings of the field in a unified and self-contained manner.

Distributionally robust optimization (DRO) is a worst-case framework for stochastic optimization under uncertainty that has drawn fast-growing studies in recent years. When the underlying probability distribution is unknown and observed from data, DRO suggests computing the worst-case distribution within a so-called uncertainty set that captures the involved statistical uncertainty. In particular, DRO with uncertainty set constructed as a statistical divergence neighborhood ball has been shown to provide a tool for constructing valid confidence intervals for nonparametric functionals and bears a duality with the empirical likelihood (EL). In this paper, we show how adjusting the ball size of such type of DRO can reduce higher-order coverage errors similar to the so-called Bartlett correction. Our correction, which applies to general von Mises differentiable functionals, is more general than the existing EL literature that only focuses on smooth function models or M-estimation. Moreover, we demonstrate a higher-order “self-normalizing” property of DRO regardless of the choice of divergence. Our approach builds on the development of a higher-order expansion of DRO, which is obtained through an asymptotic analysis on a fixed-point equation arising from the Karush-Kuhn-Tucker conditions. Funding: This work was supported by the National Science Foundation, Division of Information and Intelligent Systems [Grant IIS-1849280] and the Division of Civil, Mechanical and Manufacturing Innovation [Grant CAREER CMMI-1834710].

Abstract In practical industrial applications, the collection of bearing fault data faces challenges such as unknown operating conditions and complex fault types. Moreover, discrepancies in feature distributions across different operating conditions hinder the effective generalization of knowledge from training domains to target domains in intelligent diagnostic models. To address these challenges, this paper proposes a novel domain generalization (DG) network for bearing fault diagnosis under unknown operating conditions. Specifically, a distributionally robust optimization (DRO) approach is integrated into the DG framework. By exploring uncertainty sets of fault features and utilizing the feature distribution of subset collections, the proposed method bridges the distribution gap between the training and unknown domains. Additionally, a distribution exploration strategy and a Wasserstein distance constraint are introduced to refine the DRO approach. These enhancements implicitly guide feature distributions toward the unknown domain by narrowing the uncertainty set’s exploration range, thereby improving model robustness. Experimental validation on the Case Western Reserve University and Huazhong University of Science and Technology bearing vibration datasets demonstrates that the proposed model outperforms several state-of-the-art fault diagnosis methods in terms of feature distribution alignment and fault type classification, achieving superior diagnostic performance under unknown operating conditions.

In distributed learning systems, robustness threat may arise from two major sources. On the one hand, due to distributional shifts between training data and test data, the trained model could exhibit poor out-of-sample performance. On the other hand, a portion of working nodes might be subject to Byzantine attacks, which could invalidate the learning result. In this article, we propose a new research direction that jointly considers distributional shifts and Byzantine attacks. We illuminate the major challenges in addressing these two issues simultaneously. Accordingly, we design a new algorithm that equips distributed learning with both distributional robustness and Byzantine robustness. Our algorithm is built on recent advances in distributionally robust optimization (DRO) as well as norm-based screening (NBS), a robust aggregation scheme against Byzantine attacks. We provide convergence proofs in three cases of the learning model being nonconvex, convex, and strongly convex for the proposed algorithm, shedding light on its convergence behaviors and endurability against Byzantine attacks. In particular, we deduce that any algorithm employing NBS (including ours) cannot converge when the percentage of Byzantine nodes is $(1/3)$ or higher, instead of $(1/2)$ , which is the common belief in current literature. The experimental results verify our theoretical findings (on the breakpoint of NBS and others) and also demonstrate the effectiveness of our algorithm against both robustness issues, justifying our choice of NBS over other widely used robust aggregation schemes. To the best of our knowledge, this is the first work to address distributional shifts and Byzantine attacks simultaneously.

Problem definition: This paper addresses an appointment scheduling problem involving multiple sequential servers using a distributionally robust optimization (DRO) approach. Two decisions are optimized: the schedule for patient visits and an adjusting policy to rebalance customers’ waiting time across servers. Methodology/results: We formulate the distributionally robust appointment scheduling problem in sequential-servers systems using conic optimization, incorporating service time correlations. We find that the traditional cost-minimization approach results in imbalanced waiting times, concentrated at downstream servers. To address this, we propose strategically idling upstream servers, inspired by queueing literature, and develop a DRO model to jointly optimize the schedule and strategic idling (SI) policy. Through extensive numerical studies, we thoroughly examine the role of SI in a system’s performance and the effect of correlation information on the optimal schedule and SI policy. Finally, using data from a clinic, we conduct a case study to demonstrate the performance advantage of our SI policy, over existing SI policies in the literature. Managerial implications: First, our SI model provides a jointly optimal schedule and SI policy that is effective in balancing waiting times across servers, in some scenarios, and also reduces total waiting time, with the cost borne by the service provider’s overtime. Second, the typical dome-shaped schedule provides an advantage in systems with multiple congested servers, as it evenly distributes congestion across servers. Finally, incorporating correlations lessens the expected cost and improves the patients’ waiting experience with balanced waiting time across servers. Funding: Z. Yan received financial support from the Ministry of Education, Singapore [AcRF Tier 1 Grant RG20/23] and the School of Physical and Mathematical Sciences Collaborative Research Award (2024). Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0278 .

Problem definition: Distributionally robust optimization (DRO) is ubiquitous to address uncertainties inherent in operations management (OM) problems. Recently, an alternative goal-driven framework, robust satisficing (RS), is proposed. RS aims to attain a prescribed target, such as avoiding overshooting the cost budget, as much as possible under uncertainty. The goal-driven modeling philosophy fits many OM problems, yet there is a lack of direct comparisons between DRO and RS. In this paper, we uncover connections between DRO and RS. Methodology/results: Suppose both models are based on the Wasserstein metric and consider a risk-aware convex objective function affected by uncertain parameters. We demonstrate that they share the same solution family. We establish the correspondence between the radius parameter in DRO and the target parameter in RS such that the optimal solutions to the two models coincide. Inspired by the globalized distributionally robust counterpart (GDRC), we extend the analysis to GDRC and the globalized robust satisficing (GRS). We reveal that GDRC and GRS have the same solution families as DRO and RS, respectively. More importantly, we establish novel results on the equivalence of DRO, GDRC, RS, and GRS models under previously stated conditions. Managerial implications: The equivalence results help unify performance bounds of DRO and RS models. Specifically, each model now has an additional set of theoretical guarantees from the other model, and any bounds derived for one model automatically apply to other equivalent models via some parameter mapping. Despite the theoretical equivalence result, the performance of the DRO and RS models can vary depending on how the model parameters are selected. The experimental findings show how these differences emerge when transitioning from theory to practice. Additionally, the experiments provide insights for practitioners, such as how the use of cross-validation can help reflect the true model preferences, particularly when only a few validation points are set. Funding: The research of Z. Wang and L. Ran was supported by the National Natural Science Foundation of China [Grants 72272014, 91746210, and 72061127001]. Z. Wang’s research was also supported by the National Natural Science Foundation of China [Grant 72242106]. The research of M. Zhou was supported by the National Natural Science Foundation of China [Grants 72301075, and 72293564/72293560]. Supplemental Material: The online appendices are available at https://doi.org/10.1287/msom.2023.0531 .

To effectively account for the impact of fluctuations in the power generation efficiency of renewable energy sources such as photovoltaics (PVs) and wind turbines (WTs), as well as the uncertainties in load demand within an integrated energy system (IES), this article develops an IES model incorporating power generation units such as PV, WT, microturbines (MTs), Electrolyzer (EL), and Hydrogen Fuel Cell (HFC), along with energy storage components including batteries and heating storage systems. Furthermore, a demand response (DR) mechanism is introduced to dynamically regulate the energy supply–demand balance. In modeling uncertainties, this article utilizes historical data on PV, WT, and loads, combined with the adjustability of decision variables, to generate a large set of initial scenarios through the Monte Carlo (MC) sampling algorithm. These scenarios are subsequently reduced using a combination of the K-means clustering algorithm and the Simultaneous Backward Reduction (SBR) technique to obtain representative scenarios. To further manage uncertainties, a distributionally robust optimization (DRO) approach is introduced. This method uses 1-norm and ∞-norm constraints to define an ambiguity set of probability distributions, thereby restricting the fluctuation range of probability distributions, mitigating the impact of deviations on optimization results, and achieving a balance between robustness and economic efficiency in the optimization process. Finally, the model is solved using the column and constraint generation algorithm, and its robustness and effectiveness are validated through case studies. The MC sampling method adopted in this article, compared to Latin hypercube sampling followed by clustering-based scenario reduction, achieves a maximum reduction of approximately 17.81% in total system cost. Additionally, the results confirm that as the number of generated scenarios increases, the optimized cost decreases, with a maximum reduction of 1.14%. Furthermore, a comprehensive cost analysis of different uncertainties modeling approaches is conducted, demonstrating that the optimization results lie between those obtained from stochastic optimization (SO) and robust optimization (RO), effectively balancing conservatism and economic efficiency.

This paper proposes a fault self-healing recovery strategy for passive low-voltage power station areas (LVPSAs). Firstly, being aware of the typical structure and communication conditions of the LVPSAs, a fog computing load forecasting method is proposed based on a dynamic aggregation of incremental learning models. This forecasting method embeds two weighted ultra-short-term load forecasting techniques of complementary characteristics and mines real-time load to learn incrementally, and thanks to this mechanism, the method can efficiently make predictions of low-voltage loads with trivial computational burden and data storage. Secondly, the low-voltage power restoration problem is overall formulated as a three-stage mixed integer program. Specifically, the master problem is essentially a mixed integer linear program, which is mainly intended for determining the reconfiguration of binary switch states, while the slave problem, aiming at minimizing load curtailment constrained by power flow balance along with inevitable load forecast errors, is cast as mixed integer type-1 Wasserstein distributionally robust optimization. The column-and-constraint generation technique is employed to expedite the model-resolving process after the slave problem with integer variables eliminated is equated with the Karush–Kuhn–Tucker conditions. Comparative case studies are conducted to demonstrate the performance of the proposed method.

Loss functions play a pivotal role in optimizing recommendation models. Among various loss functions, Softmax Loss (SL) and Cosine Contrastive Loss (CCL) are particularly effective. Their theoretical connections and differences warrant in-depth exploration. This work conducts comprehensive analyses of these losses, yielding significant insights: 1) Common strengths --- both can be viewed as augmentations of traditional losses with Distributional Robust Optimization (DRO), enhancing robustness to distributional shifts; 2) Respective limitations --- stemming from their use of different distribution distance metrics in DRO optimization, SL exhibits high sensitivity to false negative instances, whereas CCL suffers from low data utilization. To address these limitations, this work proposes a new loss function, DrRL, which generalizes SL and CCL by leveraging Rényi-divergence in DRO optimization. DrRL incorporates the advantageous structures of both SL and CCL, and can be demonstrated to effectively mitigate their limitations. Extensive experiments have been conducted to validate the superiority of DrRL on both recommendation accuracy and robustness.

The rapid adoption of electric vehicles (EVs) and the increasing reliance on renewable energy sources necessitate innovative charging infrastructure solutions to address key challenges in energy efficiency, grid stability, and sustainable transportation. Dynamic wireless charging systems, which enable EVs to charge while in motion, offer a transformative approach to mitigating range anxiety and optimizing energy utilization. However, these systems face significant operational challenges, including dynamic traffic conditions, uncertain EV arrival patterns, energy transfer efficiency variations, and renewable energy intermittency. This paper proposes a novel quantum computing-assisted optimization framework for the modeling, operation, and control of wireless dynamic charging infrastructure in urban highway networks. Specifically, we leverage Variational Quantum Algorithms (VQAs) to address the high-dimensional, multi-objective optimization problem associated with real-time energy dispatch, charging pad utilization, and traffic flow coordination. The mathematical modeling framework captures critical aspects of the system, including power balance constraints, state-of-charge (SOC) dynamics, stochastic vehicle arrivals, and charging efficiency degradation due to vehicle misalignment and speed variations. The proposed methodology integrates quantum-inspired optimization techniques with classical distributionally robust optimization (DRO) principles, ensuring adaptability to system uncertainties while maintaining computational efficiency. A comprehensive case study is conducted on a 50 km urban highway network equipped with 20 charging pad segments, supporting an average traffic flow of 10,000 EVs per day. The results demonstrate that the proposed quantum-assisted approach significantly enhances energy efficiency, reducing energy losses by up to 18% compared to classical optimization methods. Moreover, traffic-aware adaptive charging strategies improve SOC recovery by 25% during peak congestion periods while ensuring equitable energy allocation among different vehicle types. The framework also facilitates a 30% increase in renewable energy utilization, aligning energy dispatch with periods of high solar and wind generation. Key insights from the case study highlight the critical impact of vehicle alignment, speed variations, and congestion on wireless charging performance, emphasizing the need for intelligent scheduling and real-time optimization. The findings contribute to advancing the integration of quantum computing into sustainable transportation planning, offering a scalable and robust solution for next-generation EV charging infrastructure.