Distributionally Robust Optimization Research Articles

Distributionally robust planning for power distribution network considering multi-energy station enabled integrated demand response

—To copy with the peak valley difference and peak load problem of power distribution network (PDN), this paper proposes a distributionally robust planning model for PDN considering the flexible management via the multi-energy station (MES) based integrated demand response (IDR). The planning covers reinforcement of the existing substation and feeders as well as the investment of MES and IDR. The IDR involves the incentive interruptible load response (ILR), transferable load response (TLR) and price-based power-gas substitution (PGS) which is implemented on MES. The MES integrates multiple energy storages and energy coupling devices, which is served as a platform for PDN operator to centrally track the final demand and improve the response potential of IDR. The proposed planning method aims at minimizing the present value of PDN investment cost and operation cost during the planning period. To handle the uncertainty of PDN end-use loads, a two-stage Kullback–Leibler divergence (KLD)-based distributionally robust optimization (DRO) planning model is formulated. Finally, the columns and constraints generation algorithm is utilized to solve the KLD-DRO planning model. Case studies are carried out on a practical 152-bus urban distribution system to validate the effectiveness of the proposed methodology and quantitatively analyze the key influence factors of PDN investment.

A Short and General Duality Proof for Wasserstein Distributionally Robust Optimization

Wasserstein distributionally robust optimization has emerged as a recent topic with broader applications in operations research and machine learning. Various proofs have been presented in the literature, each differing in assumptions and levels of generality. In “A Short and General Duality Proof for Wasserstein Distributionally Robust Optimization,” Zhang, Yang, and Gao present a novel elementary proof that not only shortens existing frameworks but also offers surprising generalizations. Leveraging classical Legendre—Fenchel duality, they demonstrate that strong duality is contingent on a certain interchangeability principle. Moreover, they extend this duality result to encompass risk-averse optimization and globalized distributionally robust counterparts.

Human-safe and economic operation of renewable hydrogen-based microgrids under plateau conditions

The cold and hypoxic environment threatens the health and quality of people's lives in plateau areas. Appropriate air pressure, oxygen, and temperature are crucial for human-safe aspects in plateau microgrids. Facing extreme scenarios in these areas, we propose a two-stage robust operation strategy for renewable hydrogen-based microgrids considering the reliable supply of hydrogen and oxygen. To cope with rapid weather changes in these areas, the distributionally robust optimization (DRO) model that incorporates first-moment information is employed to depict renewable energy uncertainty. This study focuses on two weakly connected microgrids with distinct characteristics, showcasing their potential for mutual support. In the first stage, the plateau microgrid fulfills demands for electric power, hydrogen, and oxygen, with the power support from the another microgrid. In the second stage, reserves such as hydrogen fuel cells, diesel generators, and hydropower are utilized to mitigate renewable energy uncertainty. The operation optimization problem based on DRO is solved using the alternating direction method of multipliers. Case studies show that the strategy can coordinate complementary operation to enhance reliability, guarantee human safety, and save costs by 39.29 % and 38.73 % for the two microgrids.

Distributionally robust origin–destination demand estimation

Gaining a good understanding of the travel demands of a city or region is extremely important for many transportation applications. For stochastic origin–destination (OD) estimation problems, an accurate distribution assumption or observation of OD estimates or data is usually desired but not always available. In this paper, we establish a novel two-stage OD estimation framework based on distributionally robust optimization (DRO) and quasi-sparsity property of large-scale OD demand matrices. The proposed two-stage Distributionally Robust Quasi-Sparsity OD estimation (DR-QSOD) model does not require an accurate or complete distribution assumption of estimates/data. Numerical results demonstrate that DR-QSOD model outperforms stochastic QSOD model in estimating OD demands when the distribution assumption of data is biased. This paper also discusses two different approaches to solve the DR-QSOD model as well as compares their computational efficiency. In addition, DR-QSOD model is shown to keep relatively high quasi-sparsity consistency, which also brings lots of meaningful practical insights.

Pre-disaster evacuation network design with uncertain demand and behavioural choices: A bi-criteria generalized cost perspective

Pre-disaster evacuation network design is critical to improving evacuation efficiency and reducing the loss of life and property caused by disasters. However, evacuation network design is complicated by the uncertainty of pre-disaster evacuation demand and the heterogeneity of evacuees’ route choices when faced with a disaster scenario. Therefore, how to design efficient evacuation networks considering these uncertainties has become an important research topic. In this paper, we study the pre-disaster flood evacuation network design problem, define a bi-criteria generalized cost metric based on Evacuation Travel Time (ETT) and Evacuation Network Congestion Degree (ENCD), and develop a model that considers uncertain demand and evacuation behavioural choices to minimize the generalized cost in the evacuation network. For the uncertain behavioural choice, we propose an Expanded Stochastic User Equilibrium (ESUE) model based on the Random Regret Minimization (RRM) principle and Boundedly Rational User Equilibrium (BRUE) conditions to describe it. For the uncertain demand, we design a Distributionally Robust Optimization (DRO) method based on Mean Absolute Deviation (MAD) to deal with this uncertain model. This paper designs an Improved Method of Successive Averages Combined with Genetic Algorithm (IMSA-GA) to solve the model and uses the Nguyen-Dupuis test network to verify the applicability and feasibility of the model and algorithm. In addition, we also conduct an evacuation network design in the Hawkesbury-Nepean region of Australia. The larger the threshold value based on the regret utility ratio, the smaller the generalized cost value, which indicates the behavioural choices of the evacuees have an impact on the evacuation network design, and the generalized cost decreases by 6.12% when the threshold value is increased from 0.1 to 0.9. In addition, the generalized cost value obtained by IMSA-GA will be 0.0025 lower than Improved Method of Successive Averages Combined with Simulated Annealing (IMSA-SA) and 0.0005 lower than Improved Method of Successive Averages Combined with Taboo Search (IMSA-TS). The theoretical contribution of this paper is to find an approach to the uncertain evacuation network design problem, especially to improve the practicality of the model by fully considering evacuation behaviour. The practical contribution is that our findings can provide insights for government authorities to design efficient and reliable evacuation networks that balance evacuation travel time and evacuation network congestion. In the future, if evacuation behavioural choices can be analysed based on actual surveys, the estimation of behavioural parameters based on survey data will be more instructive for the evacuation process.

Data-driven aerodynamic shape design with distributionally robust optimization approaches

We formulate and solve data-driven aerodynamic shape design problems with distributionally robust optimization (DRO) approaches. DRO aims to minimize the worst-case expected performance in a set of distributions that is informed by observed data with uncertainties. Building on the findings of the work Gotoh, et al. (2018), we study the connections between a class of DRO and robust design optimization, which is classically based on the mean–variance (standard deviation) optimization formulation pioneered by Taguchi. Our results provide a new perspective to the understanding and formulation of robust design optimization problems. It enables data-driven and statistically principled approaches to quantify the trade-offs between robustness and performance, in contrast to the classical robust design formulation that captures uncertainty only qualitatively. Our preliminary computational experiments on aerodynamic shape optimization in transonic turbulent flow show promising design results.

Integrated scheduling of multiple heterogeneous equipment and maintenance operations during discharging process in a container terminal under uncertainty

The integrated scheduling problem of container-handling equipment has been gaining prestige in recent years instead of decomposing the problem into a succession of subordinate sub-problems. This is due to decreasing operational costs, waiting time, completion time, and increasing efficiency. This study introduces an unprecedented integrated scheduling problem in a container terminal, wherein a broad range of heterogeneous container-handling equipment are considered, including gantry cranes, straddle carriers, transtainers, trailers, and outbound trucks. In this regard, an optimization model is developed for planning the discharging of heterogeneous containers, including storage and clearance, in three distinctive operational yards, including quay, storage, and clearance. Additionally, a wide range of realistic concerns are reflected, including maintenance operations, transport efficiency, and non-interference constraints. Furthermore, since maintenance operations’ times undeniably impact scheduling programming and involve various uncertainties, a new two-stage data-driven approach is offered to estimate reliable maintenance times. In the first stage, a hybrid machine learning approach is devised to estimate maintenance times, including the improved complete ensemble empirical mode decomposition with adaptive noise (ICEEMDAN), the Takagi–Sugeno–Kang (TSK) fuzzy system, and the minibatch gradient descent with regularization, DropRule, and AdaBound (MBGD-RDA). In the second stage, since estimated maintenance operation times have uncertainty, the distributionally robust optimization (DRO) method with φ-divergence is utilized to tackle this challenge. Finally, a case study and various simulation experiments are investigated to illustrate the validation and applicability of the suggested framework.

Joint optimization of dynamic pricing, quality, and production quantity in the presence of online reviews: A distributionally robust optimization approach

This paper investigates the joint pricing, product quality, and production quantity decisions of a firm in a two-stage scenario, considering potential market demand uncertainty and the existence of online reviews. Two decision models are established based on whether the firm adjusts product quality and prices according to consumer feedback from reviews: a static model (as a baseline) and a dynamic model. In the static model, the firm decides product quality, prices, and production quantity in the first stage, while in the second stage, the firm does not enhance product quality, and prices remain unchanged. In the dynamic model, in the first stage, the firm decides initial product quality and pricing, then in the second stage, adjusts product quality based on consumer feedback and sets pricing accordingly. We propose a distributionally robust optimization (DRO) method to address this problem, obtaining a closed-form solution for the expected profit function under the worst distribution, making it easy to determine the optimal decision. Through numerical simulation analysis, we find that the firm implementing dynamic decisions always make higher profits compared to those using static decisions. This highlights the superiority of dynamic decisions in adapting to market changes and improving product quality. We also analyze the impact of expected and standard deviation of potential market demand, as well as consumer private assessment weight, on business decisions and profits. Our research provides effective support for firms with limited market information, enabling them to better formulate joint pricing, quality, and production decisions.

Distributionally robust optimization configuration method for island microgrid considering extreme scenarios

The marine climate conditions are intricate and variable. In scenarios characterized by high proportions of wind and solar energy access, the uncertainty regarding the energy sources for island microgrid is significantly exacerbated, presenting challenges to both the economic viability and reliability of the capacity configuration for island microgrids. To address this issue, this paper proposes a distributionally robust optimization (DRO) method for island microgrids, considering extreme scenarios of wind and solar conditions. Firstly, to address the challenge of determining the probability distribution functions of wind and solar in complex island climates, a conditional generative adversarial network (CGAN) is employed to generate a scenario set for wind and solar conditions. Then, by combining k-means clustering with an extreme scenario selection method, typical scenarios and extreme scenarios are selected from the generated scenario set, forming the scenario set for the DRO model of island microgrids. On this basis, a DRO model based on multiple discrete scenarios is constructed with the objective of minimizing the sum of investment costs, operation and maintenance costs, fuel purchase costs, penalty costs of wind and solar curtailment, and penalty costs of load loss. The model is subjected to equipment operation and power balance constraints, and solved using the columns and constraints generation (CCG) algorithm. Finally, through typical examples, the effectiveness of this paper’s method in balancing the economic viability and robustness of the configuration scheme for the island microgrid, as well as reducing wind and solar curtailment and load loss, is verified.

Distributionally robust optimal dispatching method for integrated energy system with concentrating solar power plant

Concentrating solar power (CSP) plants have significant potential to complement the growing wind energy in power scheduling. This study examines an integrated energy system (IES) that incorporates a wind turbine (WT), CSP, and combined heat and power (CHP) to promote the utilization of renewable energy (RE), reduce fluctuations caused by uncertainty, and enhance the economic viability of the system. We propose a distributionally robust optimization (DRO) model for IES scheduling that considers the uncertainty of wind power by using an ambiguity set defined by the Wasserstein metric. Before the occurrence of uncertainties, the system determines the initial dispatching scheme based on the forecast data. In the second stage, the system aims to minimize the adjustment cost expectation under the worst distribution of the ambiguity set and adjusts the flexible resources in real time to offset the fluctuations caused by forecasting errors. The proposed DRO model is transformed into a conventional two-stage robust problem using strong dual theory and KKT conditions, and then solved with a modified column-and-constraint generation (C&CG) algorithm. The results of case studies show that the CSP plant enhances the system's flexibility and controllability through thermal energy storage (TES).

Stochastic First-Order Algorithms for Constrained Distributionally Robust Optimization

We consider distributionally robust optimization (DRO) problems, reformulated as distributionally robust feasibility (DRF) problems, with multiple expectation constraints. We propose a generic stochastic first-order meta-algorithm, where the decision variables and uncertain distribution parameters are each updated separately by applying stochastic first-order methods. We then specialize our results to the case of using two specific versions of stochastic mirror descent (SMD): (i) a novel approximate version of SMD to update the decision variables, and (ii) the bandit mirror descent method to update the distribution parameters in the case of [Formula: see text]-divergence sets. For this specialization, we demonstrate that the total number of iterations is independent of the dimensions of the decision variables and distribution parameters. Moreover, the cost per iteration to update both sets of variables is nearly independent of the dimension of the distribution parameters, allowing for high-dimensional ambiguity sets. Furthermore, we show that the total number of iterations of our algorithm has a logarithmic dependence on the number of constraints. Experiments on logistic regression with fairness constraints, personalized parameter selection in a social network, and the multi-item newsvendor problem verify the theoretical results and show the usefulness of the algorithm, in particular when the dimension of the distribution parameters is large. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms—Continuous. Funding: This work was supported by the National Science Foundation [Grant 2112533]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2023.0167 .

Novel integer L-shaped method for parallel machine scheduling problem under uncertain sequence-dependent setups

We study scheduling problems on unrelated parallel machines with uncertainty in job processing and sequence-dependent setup times. We first formulate this problem as a two-stage stochastic program using a scenario-based approach, in which the first-stage decisions involve job-machine assignments, and the second-stage deals with job sequencing. We then propose an innovative modified integer L-shaped method, uniquely designed to decompose the primary problem into independent subproblems at both the scenario and machine levels. We have augmented this method with a specialized integer optimality cut, which is further strengthened using a sequential lifting procedure. To broaden the applicability of our model and solution, we have adapted it to a distributionally robust optimization (DRO) framework under Wasserstein ambiguity. This process involves reformulating the DRO model into a mixed-integer linear program (MILP) using conic duality theory. Additionally, we integrate a distribution separation program to define the worst-case probability distribution during each iteration of the proposed integer L-shaped method, enhancing the computational speed for solving the DRO model. We conducted a series of numerical experiments to illustrate the effectiveness and scalability of the proposed solution methodology, which demonstrated highly promising results. Our innovative decomposition algorithms showed superior performance, achieving solution times that are significantly faster (up to two orders of magnitude) compared to the traditional integer L-shaped method for the stochastic scheduling model and the direct MILP reformulation of the DRO model. These results were obtained for problem instances solved to the exactness within the time limit. Moreover, the solution derived from the DRO model exhibited greater robustness in out-of-sample analysis with simulated data compared to the solution obtained from the stochastic model.

EXPRESS: Fairness as a Robust Utilitarianism

This article focuses on the social choice problem in which decisions are based on the utility of multiple stakeholder types. The sum of these utilities – Utilitarian welfare - is one of the most important objectives in solving the social choice problem. While it is the most efficient solution, maximizing Utilitarian welfare may lead to unfair outcomes. However, Encouraging a Utilitarian decision-maker to adopt a fair decision is challenging due to the associated efficiency loss. This article takes a novel perspective by motivating a Utilitarian decision-maker to make fair decisions from an uncertainty-averse standpoint. We study the problem where the proportions of stakeholder types are uncertain and propose a distributionally robust optimization (DRO) model that maximizes the worst-case Utilitarian welfare over an ϕ-divergence-based uncertainty set. We provide three aspects of the relationship between fairness and the uncertain-averse Utilitarian welfare maximization. First, we establish that the worst-case Utilitarian welfare adheres to all five axioms of unfairness-averse cardinal welfare functions with two stakeholder types and satisfies four of these when this number exceeds two. Second, we demonstrate that with the maximal extent of uncertainty aversion, the DRO model identifies the Egalitarian welfare maximizer, which prioritizes fairness. Further, given serveral conventional assumptions, the proposed model selects the Nash welfare maximizer, an objective trade-off between efficiency and fairness, with moderate levels of uncertainty aversion. Lastly, we present numerical studies of two specific instances of the social choice problem – resource allocation and facility location – to show that, as uncertainty aversion increases, our model increasingly emphasizes fairness.

Enhancing Efficiency and Energy Optimization: Data-Driven Solutions in Process Industrial Manufacturing

This paper reviews the current state of research in data analytics and machine learning techniques, focusing on their applications in process industrial manufacturing, particularly in control and optimization. Key areas for future research include selection and transfer learning for process monitoring, addressing time-varying characteristics, and enhancing data-driven optimal control with domain-specific knowledge. Additionally, the paper explores reinforcement learning techniques and robust optimization, including distributional robust optimization, for high-level decision-making. Emphasizing the importance of historical knowledge of plants and processes, this paper aims to identify knowledge gaps and pave the way for future research in data-driven strategies for process industries, with a particular emphasis on energy efficiency and optimization.

Sizing Energy Storage Systems to Dispatch Wind Power Plants

Integrating wind power plants into the electricity grid poses challenges due to the intermittent nature of wind energy generation. Energy storage systems (ESSs) have shown promise in mitigating the intermittent variability associated with wind power. This paper presents a distributionally robust optimization (DRO) model for sizing energy storage systems to dispatch wind power plants. The variable wind power is formulated as a moment-based ambiguity set. Dispatchability is described by the expected value of the insufficient power of wind power relative to the dispatch command, which is a sum of nonlinear functions and is taken as the optimal index. A deterministic semi-definite positive model is derived to solve the problem effectively. Numerical studies are conducted to demonstrate the effectiveness and advantages of the proposed method.

Low-carbon economic scheduling for AIES considering flexible load coordination and multiple uncertainties

The disordered energy use in traditional agricultural production has resulted in high energy purchase costs and carbon emissions. The coupling of integrated energy systems with agriculture offers the potential for flexible coordination of agricultural loads while agricultural loads are highly influenced by changes in the natural environment. In this context, this paper proposes a low-carbon and economic scheduling optimization method for agricultural integrated energy system (AIES), which comprehensively takes temperature, light intensity, and the coordination of agricultural loads into consideration. Firstly, solar and biomass resources are integrated through energy conversion equipment to support energy supply from both the electrical distribution network (EDN) and the gas network. At the same time, the modelling process of agricultural load is refined, which is divided into crop irrigation, greenhouse temperature control and residential consumption. The models of agricultural multi-energy complementary and energy storage equipment are established to achieve flexible conversion between different energy sources. In addition, the distributionally robust optimization (DRO) method based on multiple discrete scenarios is adopted to deal with the uncertainties arising from environmental condition fluctuations during the optimization process. Finally, the proposed model is tested on 4 various regions and 2 different seasons. Numerical results show that the proposed model can effectively enhance the economic and environmental performance of the AIES.

Data-driven distributionally robust day-ahead dispatch for active distribution networks based on improved conditional generative adversarial network

With the increasing penetration of renewable energy generation (REG), strong uncertainty poses a significant challenge to the rationality of day-ahead dispatch plans in distribution systems. We propose a data-driven, distributionally robust day-ahead dispatch method for active distribution networks (ADNs) based on an improved conditional generative adversarial network (CGAN). First, an improved CWGAN-GP model is constructed based on three-dimensional convolution (Conv3D) and the Wasserstein distance to obtain the day-ahead REG scenarios, considering the correlation features of photovoltaic (PV) and wind turbines (WTs), which reduces the conservatism of scenario generation. Accordingly, the Gaussian mixture model (GMM) clustering method based on moment information was used to construct an ambiguity set. Moreover, a two-stage distributionally robust day-ahead dispatch model for an ADN is established, which realizes the fusion of the data-driven method and distributionally robust optimization (DRO) mechanism model. Finally, the validity of the data-driven DRO (DDRO) method is verified through a case analysis.

Learning With Noisy Labels Over Imbalanced Subpopulations.

Learning with noisy labels (LNL) has attracted significant attention from the research community. Many recent LNL methods rely on the assumption that clean samples tend to have a "small loss." However, this assumption often fails to generalize to some real-world cases with imbalanced subpopulations, that is, training subpopulations that vary in sample size or recognition difficulty. Therefore, recent LNL methods face the risk of misclassifying those "informative" samples (e.g., hard samples or samples in the tail subpopulations) into noisy samples, leading to poor generalization performance. To address this issue, we propose a novel LNL method to deal with noisy labels and imbalanced subpopulations simultaneously. It first leverages sample correlation to estimate samples' clean probabilities for label correction and then utilizes corrected labels for distributionally robust optimization (DRO) to further improve the robustness. Specifically, in contrast to previous works using classification loss as the selection criterion, we introduce a feature-based metric that takes the sample correlation into account for estimating samples' clean probabilities. Then, we refurbish the noisy labels using the estimated clean probabilities and the pseudo-labels from the model's predictions. With refurbished labels, we use DRO to train the model to be robust to subpopulation imbalance. Extensive experiments on a wide range of benchmarks demonstrate that our technique can consistently improve state-of-the-art (SOTA) robust learning paradigms against noisy labels, especially when encountering imbalanced subpopulations. We provide our code in https://github.com/chenmc1996/LNL-IS.

Wasserstein Robust Classification with Fairness Constraints

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-[Formula: see text] 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 .

Ship coping strategies for hurricane-induced port disruptions

PurposeDisruptions at ports may destroy the planned ship schedules profoundly, which is an imperative operation problem that shipping companies need to overcome. This paper attempts to help shipping companies cope with port disruptions through recovery scheduling.Design/methodology/approachThis paper studies the ship coping strategies for the port disruptions caused by severe weather. A novel mixed-integer nonlinear programming model is proposed to solve the ship schedule recovery problem (SSRP). A distributionally robust mean conditional value-at-risk (CVaR) optimization model was constructed to handle the SSRP with port disruption uncertainties, for which we derive tractable counterparts under the polyhedral ambiguity sets.FindingsThe results show that the size of ambiguity set, confidence level and risk-aversion parameter can significantly affect the optimal values, decision-makers should choose a reasonable parameter combination. Besides, sailing speed adjustment and handling rate adjustment are effective strategies in SSRP but may not be sufficient to recover the schedule; therefore, port skipping and swapping are necessary when multiple or longer disruptions occur at ports.Originality/valueSince the port disruption is difficult to forecast, we attempt to take the uncertainties into account to achieve more meaningful results. To the best of our knowledge, there is barely a research study focusing on the uncertain port disruptions in the SSRP. Moreover, this is the first paper that applies distributionally robust optimization (DRO) to deal with uncertain port disruptions through the equivalent counterpart of DRO with polyhedral ambiguity set, in which a robust mean-CVaR optimization formulation is adopted as the objective function for a trade-off between the expected total costs and the risk.