Uncertainty Set Research Articles

Solving Non-rectangular Reward-Robust MDPs via Frequency Regularization

In robust Markov decision processes (RMDPs), it is assumed that the reward and the transition dynamics lie in a given uncertainty set. By targeting maximal return under the most adversarial model from that set, RMDPs address performance sensitivity to misspecified environments. Yet, to preserve computational tractability, the uncertainty set is traditionally independently structured for each state. This so-called rectangularity condition is solely motivated by computational concerns. As a result, it lacks a practical incentive and may lead to overly conservative behavior. In this work, we study coupled reward RMDPs where the transition kernel is fixed, but the reward function lies within an alpha-radius from a nominal one. We draw a direct connection between this type of non-rectangular reward-RMDPs and applying policy visitation frequency regularization. We introduce a policy-gradient method, and prove its convergence. Numerical experiments illustrate the learned policy's robustness and its less conservative behavior when compared to rectangular uncertainty.

Large-Scale Non-convex Stochastic Constrained Distributionally Robust Optimization

Distributionally robust optimization (DRO) is a powerful framework for training robust models against data distribution shifts. This paper focuses on constrained DRO, which has an explicit characterization of the robustness level. Existing studies on constrained DRO mostly focus on convex loss function, and exclude the practical and challenging case with non-convex loss function, e.g., neural network. This paper develops a stochastic algorithm and its performance analysis for non-convex constrained DRO. The computational complexity of our stochastic algorithm at each iteration is independent of the overall dataset size, and thus is suitable for large-scale applications. We focus on the general Cressie-Read family divergence defined uncertainty set which includes chi^2-divergences as a special case. We prove that our algorithm finds an epsilon-stationary point with an improved computational complexity than existing methods. Our method also applies to the smoothed conditional value at risk (CVaR) DRO.

Evaluating the reliability of microgrids consisting of renewable energy sources using stochastic scheduling based on the data-driven uncertainty set

The paper addresses challenges arising from the unpredictable and probabilistic nature of wind speed and solar irradiance, leading to variable power production in renewable sources. Power imbalance issues in grids result from the difficulty in accurately estimating these variables. The study introduces an improved data-driven uncertainty set, utilizing a neural network trained with extensive historical data from a multi-microgrid system. This set efficiently identifies strong coupling and extracts characteristic scenarios, ensuring high reliability with a minimal number of scenes. The proposed probabilistic model, implemented in the coordination of pumped storage, wind, and solar power plants, incorporates energy limits and storage characteristics. A four-mode IEEE model for storage units allows quick response to system needs. Simulating storage unit states every hour, considering system load, output power, and energy limitations, optimizes power dispatch based on battery capacity. Additionally, the paper addresses the uncertainty in electric power demand in microgrids, proposing an improved data-driven uncertainty set obtained from a neural network. By eliminating unrealistic worst-case scenarios, the approach accurately captures system characteristics, maintaining high reliability while significantly reducing convergence time. The proposed uncertainty set is applied in a novel two-stage robust optimization dispatch model, transforming the optimization problem into a mixed-integer linear programming problem, and solving it using a column and constraint generation algorithm. Numerical case studies verify the feasibility and superiority of the proposed approach in terms of economy and convergence performance compared to existing robust optimization methods.

Data-driven optimization for seismic-resilient power network planning

Many regions of the planet are exposed to seismic hazards that can cause devastating consequences on power systems. Due to these systems’ crucial role, the evaluation and planning for their safe and reliable operation are paramount. This paper develops a novel data-driven optimization framework to assess the power network’s seismic resilience and plan cost-effective investments for its enhancement. Under a robust optimization scheme, an earthquake attacker–defender model finds the worst-case realization of random earthquake network contingencies within an uncertainty set defined with a large number of scenarios generated by state-of-the-art engineering methods. Moreover, data-driven stochastic-robust optimization is employed in a two-stage seismic-resilient power network planning model, leveraging multiple seismic sources’ distributional information. Transmission line expansions and siting and sizing of battery energy storage systems are decided in the first stage, while the second stage decides operational variables. Experiments on a 281-node Chilean power system provide insights for seismic-resilient planning and demonstrate the efficiency of the proposed approach.

A scenario-based robust approach for joint planning of multi-blood product logistics and multi-casualty type evacuation

This paper presents a two-stage hybrid robust programming model that integrates multiple blood products scheduling and multiple casualty types evacuation under facility disruptions risk and casualty number uncertainty. The model seeks to maximize the rescue efficiency and minimize the total operations cost. Under a set of disruption scenarios, we adopt a scenario-based robust method to address disruption risks at temporary medical centres and two robust uncertainty sets to deal with uncertain casualty numbers. We propose a proximal bundle algorithm to solve large-scale instances of the proposed hybrid robust model approximately. Extensive numerical experiments show that: (i) the trade-off between model robustness and solution robustness can help the decision-makers determine an appropriate risk-aversion weight; (ii) compared with the corresponding stochastic and scenario-based robust models, the hybrid robust model can obtain more robust solutions with a slight increase in cost; (iii) the proximal bundle algorithm can produce near-optimal solutions within reasonable computational times; (iv) some model parameters have significant impact on the total cost, which can help decision-makers set the appropriate parameters to achieve the desired outcome. Finally, we also use the real data of the 2008 Wenchuan County Earthquake in Sichuan Province, China, to illustrate the application of the model.

Two-stage robust planning method for distribution network energy storage based on load forecasting

A two-stage robust planning method for energy storage in distribution networks based on load prediction is proposed to address the uncertainty of active load in energy storage planning. First, considering the uncertainty of active load, a short-term load forecasting model combining the mutual information method and BiLSTM is established based on k-means++ clustering. Second, based on the results of load forecasting, a comprehensive norm-constrained uncertainty set is constructed, and a two-stage robust model for distribution network energy storage planning is established. The first stage aims to minimize the annual investment cost of the energy storage system, while the second stage aims to minimize the daily operating cost of the distribution network. At the same time, a second-order cone relaxation transformation model with non-convex constraints is introduced to ultimately achieve the optimal economy of the distribution network in energy storage planning. Finally, the effectiveness of the proposed method and model is validated on the IEEE 33-node distribution network model using the MATLAB platform.

Worst-case risk measures of stop-loss and limited loss random variables under distribution uncertainty with applications to robust reinsurance

Stop-loss and limited loss random variables are two important transforms of a loss random variable and appear in many modeling problems in insurance, finance, and other fields. Risk levels of a loss variable and its transforms are often measured by risk measures. When only partial information on a loss variable is available, risk measures of the loss variable and its transforms cannot be evaluated effectively. To deal with the situation of distribution uncertainty, the worst-case values of risk measures of a loss variable over an uncertainty set, describing all the possible distributions of the loss variable, have been extensively used in robust risk management for many fields. However, most of these existing results on the worst-case values of risk measures of a loss variable cannot be applied directly to the worst-case values of risk measures of its transforms. In this paper, we derive the expressions of the worst-case values of distortion risk measures of stop-loss and limited loss random variables over an uncertainty set introduced in Bernard et al. (2023). This set represents a decision maker’s belief in the distribution of a loss variable. We find the distributions under which the worst-case values are attainable. These results have potential applications in a variety of fields. To illustrate their applications, we discuss how to model optimal stop-loss reinsurance problems and how to determine optimal stop-loss retentions under distribution uncertainty. Explicit and closed-form expressions for the worst-case TVaRs of stop-loss and limited loss random variables and optimal stop-loss retentions are given under special forms of the uncertainty set. Numerical results are presented under more general forms of the uncertainty set.

Robust optimisation for vertiport location problem considering travel mode choice behaviour in urban air mobility systems

With the rapid development of novel vehicle technologies, air taxis are becoming a new transport servicing model to achieve better urban air mobility (UAM) systems. The UAM system can mitigate severe traffic congestion problems on the ground in many metropolitan areas effectively. This research aims to design a robust UAM network to serve broader mobility needs in the future. In this study, we consider the two main competition transportation modes of air taxis (luxury ground taxis and ordinary ground taxis) and establish a travel mode choice behaviour model based on the multinomial logit choice approach to determine the aggregate air taxi demand flow. Furthermore, we account for the varying temporal preferences of different UAM passengers by incorporating different time values for heterogeneous passengers, such as leisure and business passengers, into the travel mode choice behaviour model. Additionally, to address uncertainties in user demand, we introduce a robust optimisation model for vertiport location under a budget uncertainty set. This model is then reformulated as a mixed-integer linear programming model to address computational challenges. Ultimately, we conduct a series of numerical experiments to showcase the effectiveness of our proposed mathematical model. The outcomes of this research offer valuable managerial insights and implications, aiding governments and UAM operators in scientifically designing UAM networks and making strategic decisions regarding infrastructure investments.

Mechanisms for Robust Local Differential Privacy.

We consider privacy mechanisms for releasing data X=(S,U), where S is sensitive and U is non-sensitive. We introduce the robust local differential privacy (RLDP) framework, which provides strong privacy guarantees, while preserving utility. This is achieved by providing robust privacy: our mechanisms do not only provide privacy with respect to a publicly available estimate of the unknown true distribution, but also with respect to similar distributions. Such robustness mitigates the potential privacy leaks that might arise from the difference between the true distribution and the estimated one. At the same time, we mitigate the utility penalties that come with ordinary differential privacy, which involves making worst-case assumptions and dealing with extreme cases. We achieve robustness in privacy by constructing an uncertainty set based on a Rényi divergence. By analyzing the structure of this set and approximating it with a polytope, we can use robust optimization to find mechanisms with high utility. However, this relies on vertex enumeration and becomes computationally inaccessible for large input spaces. Therefore, we also introduce two low-complexity algorithms that build on existing LDP mechanisms. We evaluate the utility and robustness of the mechanisms using numerical experiments and demonstrate that our mechanisms provide robust privacy, while achieving a utility that is close to optimal.

Robust Unit Commitment With Decision-Dependent Uncertain Demand and Time-of-Use Pricing

Two-stage robust unit commitment (UC) is nowadays an effective decision-making model to support power system operations under uncertainty. This study extends the classical robust UC model to a novel formulation where the set of uncertain demand endogenously depends on the first-stage price decisions. We employ a generic time-of-use (ToU) pricing scheme, in a regulated system, to determine the optimal peak and off-peak price rates and periods of individual load buses. To exactly compute such ToU-based robust UC with decision-dependent demand, we customize and implement a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">parametric</i> variant of the column-and-constraint generation method to handle this complex program with a changing uncertainty set. The proposed model and the computational performance of the solution method are illustrated on two typical systems, and experimental results are discussed with a set of novel insights.

Robust maximum expert consensus model with adjustment path under uncertain environment

The maximum expert consensus model (MECM) is a commonly used consensus model in group decision making (GDM). In traditional MECM, the consensus constraints are not fully considered and the adjustment cost of decision maker (DM) is certain. Moreover, directing the DM’s opinion in a visual path is seldom considered in the consensus reaching process (CRP) of MECM. Inspired by these issues, this paper first proposed two MECMs with different types of consensus constraints. Then, this paper incorporated an adjustment path into MECM by using feedback coefficients that can prevent opinions from being overadjusted. Furthermore, the robust MECM (RMECM) is developed to address the uncertainty of unit adjustment cost under three uncertainty sets. Finally, the feasibility of the proposed models is verified by applying them to the allocation of special funds in the Chinese film industry, which is a large-scale group decision making (LSGDM) problem. The sensitivity analysis and comparative analysis are also conducted to show the efficiency of the proposed models.

A unified approach to inverse robust optimization problems

A variety of approaches has been developed to deal with uncertain optimization problems. Often, they start with a given set of uncertainties and then try to minimize the influence of these uncertainties. The reverse view is to first set a budget for the price one is willing to pay and then find the most robust solution. In this article, we aim to unify these inverse approaches to robustness. We provide a general problem definition and a proof of the existence of its solution. We study properties of this solution such as closedness, convexity, and boundedness. We also provide a comparison with existing robustness concepts such as the stability radius, the resilience radius, and the robust feasibility radius. We show that the general definition unifies these approaches. We conclude with an example that demonstrates the flexibility of the introduced concept.

A robust optimization approach for placement of applications in edge computing considering latency uncertainty

Edge computing brings computing and storage resources close to end-users to support new applications and services that require low network latency. It is currently used in a wide range of industries, from industrial automation and augmented reality, to smart cities and connected vehicles, where low latency, data privacy, and real-time processing are critical requirements. The latency of accessing applications in edge computing must be consistently below a threshold of a few tens of milliseconds to maintain an acceptable experience for end-users. However, the latency between users and applications can vary considerably depending on the network load and mode of wireless access. An application provider must be able to guarantee that requests are served in a timely manner by their application instances hosted in the edge despite such latency variations. This article focuses on the placement and traffic allocation problem faced by application providers in determining where to place application instances on edge nodes such that requests are served within a certain deadline. It proposes novel formulations based on robust optimization to provide optimal plans that protect against latency variations in a configurable number of network links. The robust formulations are based on two different types of polyhedral uncertainty sets that offer different levels of protection against variations in latency. Extensive simulations show that our robust models are able to keep the number of chosen edge nodes low while reducing the number of latency violations as compared to a deterministic optimization model that only considers the average latency of network links.

Supply Chain Contracts in the Small Data Regime

Problem definition: We study supply chain contract design under uncertainty. In this problem, the retailer has full information about the demand distribution, whereas the supplier only has partial information drawn from historical demand realizations and contract terms. The supplier wants to optimize the contract terms, but she only has limited data on the true demand distribution. Methodology/results: We show that the classical approach for contract design is fragile in the small data regime by identifying cases where it incurs a large loss. We then show how to combine the historical demand and retailer data to improve the supplier’s contract design. On top of this, we propose a robust contract design model where the uncertainty set requires little prior knowledge from the supplier. We show how to optimize the supplier’s worst-case profit based on this uncertainty set. In the single-product case, the worst-case profit can be found with bisection search. In the multiproduct case, the worst-case profit can be found with a cutting plane algorithm. Managerial implications: Our framework demonstrates the importance of combining the demand and retailer information into the supplier’s contract design problem. We also demonstrate the advantage of our robust model by comparing it against classical data-driven approaches. This comparison sheds light on the value of information from interactions between agents in a game-theoretic setting and suggests that such information should be utilized in data-driven decision making. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0325 .

Reliable Frequency Regulation Through Vehicle-to-Grid: Encoding Legislation with Robust Constraints

Problem definition: Vehicle-to-grid increases the low utilization rate of privately owned electric vehicles by making their batteries available to electricity grids. We formulate a robust optimization problem that maximizes a vehicle owner’s expected profit from selling primary frequency regulation to the grid and guarantees that market commitments are met at all times for all frequency deviation trajectories in a functional uncertainty set that encodes applicable legislation. Faithfully modeling the energy conversion losses during battery charging and discharging renders this optimization problem nonconvex. Methodology/results: By exploiting a total unimodularity property of the uncertainty set and an exact linear decision rule reformulation, we prove that this nonconvex robust optimization problem with functional uncertainties is equivalent to a tractable linear program. Through extensive numerical experiments using real-world data, we quantify the economic value of vehicle-to-grid and elucidate the financial incentives of vehicle owners, aggregators, equipment manufacturers, and regulators. Managerial implications: We find that the prevailing penalties for nondelivery of promised regulation power are too low to incentivize vehicle owners to honor the delivery guarantees given to grid operators. Funding: This work was supported by the Institut Vedecom. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0154 .

Designing tractable piecewise affine policies for multi-stage adjustable robust optimization

AbstractWe study piecewise affine policies for multi-stage adjustable robust optimization (ARO) problems with non-negative right-hand side uncertainty. First, we construct new dominating uncertainty sets and show how a multi-stage ARO problem can be solved efficiently with a linear program when uncertainty is replaced by these new sets. We then demonstrate how solutions for this alternative problem can be transformed into solutions for the original problem. By carefully choosing the dominating sets, we prove strong approximation bounds for our policies and extend many previously best-known bounds for the two-staged problem variant to its multi-stage counterpart. Moreover, the new bounds are—to the best of our knowledge—the first bounds shown for the general multi-stage ARO problem considered. We extensively compare our policies to other policies from the literature and prove relative performance guarantees. In two numerical experiments, we identify beneficial and disadvantageous properties for different policies and present effective adjustments to tackle the most critical disadvantages of our policies. Overall, the experiments show that our piecewise affine policies can be computed by orders of magnitude faster than affine policies, while often yielding comparable or even better results.

Climate-Adaptive Transmission Network Expansion Planning Considering Evolutions of Resources

Weather-sensitive resources are the main source of uncertainties in power systems. However, the unpredictable climate change further introduces ambiguity (i.e., unknown probability distribution) into the system, since the weather-sensitive resources would evolve with the climate and gradually exhibit a different probability distribution from the past in an uncertain manner. Lack of considering this climate-induced ambiguity in transmission network expansion planning (TNEP) may cause misunderstanding of future operational scenarios. Aiming at a higher security operation level under climate change yet less line investment, this paper proposes a climate-adaptive TNEP, which is essentially a robust TNEP equipped with a climate-adaptive uncertainty set (CUS) to embody injections from the weather-sensitive resources that have high probabilities in the target year despite the climate-induced ambiguity. Determination of the CUS involves three steps. First, model future unknown distribution under climate change. Specifically, the climate-driven evolution in distributions is quantified by an evolutionary distance between historical and future true distributions, whose upper bound is derived from practical data, while the future unknown distribution is then modeled by a distance-based ambiguity set; Second, determine the CUS which has a minimal volume yet a desired confidence level in the face of the ambiguous future distribution. To that end, a parametric Wasserstein distance-based distributionally robust optimization ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> -WDRO) is developed over the ambiguity set; Third, solve the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> -WDRO by a data-clustering-incorporated reformulation. After the CUS is determined, the overall climate-adaptive TNEP is solved by a column-and-constraint-generation method with an inner multi-loop algorithm tailored for the CUS. Simulations are conducted on three test systems with data from Australia and Coupled Model Intercomparison Project Phase 6 (CMIP6) under scenarios issued by Intergovernmental Panel on Climate Change (IPCC), which demonstrate that the climate-adaptive TNEP can improve operational security under climate change while reducing investment costs.

Adjustability in robust linear optimization

AbstractWe investigate the concept of adjustability—the difference in objective values between two types of dynamic robust optimization formulations: one where (static) decisions are made before uncertainty realization, and one where uncertainty is resolved before (adjustable) decisions. This difference reflects the value of information and decision timing in optimization under uncertainty, and is related to several other concepts such as the optimality of decision rules in robust optimization. We develop a theoretical framework to quantify adjustability based on the input data of a robust optimization problem with a linear objective, linear constraints, and fixed recourse. We make very few additional assumptions. In particular, we do not assume constraint-wise separability or parameter nonnegativity that are commonly imposed in the literature for the study of adjustability. This allows us to study important but previously under-investigated problems, such as formulations with equality constraints and problems with both upper and lower bound constraints. Based on the discovery of an interesting connection between the reformulations of the static and fully adjustable problems, our analysis gives a necessary and sufficient condition—in the form of a theorem-of-the-alternatives—for adjustability to be zero when the uncertainty set is polyhedral. Based on this sharp characterization, we provide two efficient mixed-integer optimization formulations to verify zero adjustability. Then, we develop a constructive approach to quantify adjustability when the uncertainty set is general, which results in an efficient and tight poly-time algorithm to bound adjustability. We demonstrate the efficiency and tightness via both theoretical and numerical analyses.

Approximate Kernel Learning Uncertainty Set for Robust Combinatorial Optimization

Support vector clustering (SVC) has been proposed in the literature as a data-driven approach to build uncertainty sets in robust optimization. Unfortunately, the resulting SVC-based uncertainty sets induces a large number of additional variables and constraints in the robust counterpart of mathematical formulations. We propose a two-phase method to approximate the resulting uncertainty sets and overcome these tractability issues. This method is controlled by a parameter defining a trade-off between the quality of the approximation and the complexity of the robust models formulated. We evaluate the approximation method on three distinct, well-known optimization problems. Experimental results show that the approximated uncertainty set leads to solutions that are comparable to those obtained with the classic SVC-based uncertainty set with a significant reduction of the computation time. History: Accepted by Andrea Lodi, Area Editor for Design and Analysis of Algorithms—Discrete. Funding: This work was supported by the German-French Academy for the Industry of the Future [Data-driven collaboration in Industrial Supply Chains project].

Low-Order Automatic Domain Splitting Approach for Nonlinear Uncertainty Mapping

This paper introduces a novel method for the automatic detection and handling of nonlinearities in a generic transformation. A nonlinearity index that exploits second-order Taylor expansions and polynomial bounding techniques is first introduced to estimate the Jacobian variation of a nonlinear transformation. This index is then embedded into a low-order automatic domain splitting algorithm that accurately describes the mapping of an initial uncertainty set through a generic nonlinear transformation by splitting the domain whenever nonlinearities grow above a predefined threshold. The algorithm is illustrated in the critical case of orbital uncertainty propagation, and it is coupled with a tailored merging process that limits the growth of the domains in time by recombining them when nonlinearities decrease. The low-order automatic domain splitting algorithm is then combined with Gaussian mixture models to accurately describe the propagation of a probability density function.