Uncertainty Set Research Articles

Robust Average-Reward Reinforcement Learning

Robust Markov decision processes (MDPs) aim to find a policy that optimizes the worst-case performance over an uncertainty set of MDPs. Existing studies mostly have focused on the robust MDPs under the discounted reward criterion, leaving the ones under the average-reward criterion largely unexplored. In this paper, we develop the first comprehensive and systematic study of robust average-reward MDPs, where the goal is to optimize the long-term average performance under the worst case. Our contributions are four-folds: (1) we prove the uniform convergence of the robust discounted value function to the robust average-reward function as the discount factor γ goes to 1; (2) we derive the robust average-reward Bellman equation, characterize the structure of its solution set, and prove the equivalence between solving the robust Bellman equation and finding the optimal robust policy; (3) we design robust dynamic programming algorithms, and theoretically characterize their convergence to the optimal policy; and (4) we design two model-free algorithms unitizing the multi-level Monte-Carlo approach, and prove their asymptotic convergence

Energy and reserve procurement in integrated electricity and heating system: A high-dimensional covariance matrix approach based on stochastic differential equations

The integration of stochastic renewable energy sources into energy systems presents challenges due to flexibility insufficiency and uncertainty modeling inaccuracy. To address these challenges, this paper explores a high-dimensional covariance matrix approach based on stochastic differential equations (SDEs) for energy and reserve co-dispatch in integrated electricity and heating systems. By capturing the long-term correlation of wind power forecast errors, the SDE model simultaneously models point forecasts and high-dimensional positive-definite covariance matrices, resulting in a compact uncertainty set. To ensure computational efficiency, we employ duality theory to transform the stochastic co-dispatch problem with the high-dimensional uncertainty set into deterministic convex programming. Additionally, a virtual heat storage model is introduced for the district heating network, leveraging the thermal inertia of the network to enhance reserve capacity. Simulations validate the calibration and sharpness of the proposed high-dimensional uncertainty set. The results demonstrate that co-dispatch using the SDE-based uncertainty set reduces operational costs by 4.9% compared to a set that does not consider the long-term correlation of wind power, under a 95% coverage rate. Moreover, the storage capabilities of district heating pipelines and smart buildings provide cost-free reserves, saving the operational costs of the two test systems by 2.8% and 1.3%, respectively.

Key drivers of vulnerability to rainfall flooding in New Orleans

IntroductionFuture urban stormwater flood risk is determined by the confluence of both climate-driven changes in precipitation patterns and the effectiveness of flood mitigation systems, such as urban drainage and pump systems. This is especially true in coastal cities protected by levee systems like New Orleans, where even present-day rainfall would be enough to cause serious flooding in the absence of extensive stormwater drainage and pumping. However, while the uncertainties associated with climate change have been well studied, uncertainties in infrastructure performance and operation have received less attention.MethodsWe investigated how these interrelated sets of uncertainties drive flood risk in New Orleans using a Robust Decision Making (RDM) approach. RDM is a framework for Decision Making Under Deep Uncertainty (DMDU) that leverages simulation models to facilitate exploration across many possible futures and the identification of decision-relevant scenarios. For our work, we leveraged a detailed Storm Water Management Model (SWMM) representation of the New Orleans urban stormwater management system to examine flood depths across the city when faced with different levels of future precipitation, sea-level rise, drainage pipe obstruction, and pumping system failure. We also estimated direct flood damage for each neighborhood in the city for this scenario ensemble. These damage estimates were then subjected to vulnerability analysis using scenario discovery—a technique designed to determine which combinations of uncertainties are most stressful to the system in terms of an outcome of interest (excess flood damage).ResultsOur results suggest that key drivers of vulnerability depend on geographic scale. Specifically, we find that possible climate-driven precipitation increases are the most important determinant of vulnerability at the citywide level. However, for some individual neighborhoods, infrastructure operation challenges under present day conditions are a more significant driver of vulnerability than possible climate-driven precipitation increases.

The Possibilistic Mean-Variance Model with Uncertain Possibility Distributions

The possibilistic mean–variance (MV) model is the counterpart of Markowitz’s MV model in the possibility theory. This study aims to examine the possibilistic MV model when the possibility distributions of stock returns are uncertain triangular fuzzy numbers. We define an uncertainty vector and use its ellipsoidal uncertainty set in a minimax optimization problem to model this uncertainty. We also show that this minimax optimization problem reduces to a strictly convex minimization problem. Thus, unlike the possibilistic MV model, we get diversified optimal portfolios uniquely with our approach. After laying down the theoretical points of our approach, we illustrate it with a real-world example in the literature by using a software package for convex optimization. To the best of our knowledge, this is the first paper that considers uncertain possibility distributions in the possibilistic MV model.

A machine learning approach to two-stage adaptive robust optimization

We propose an approach based on machine learning to solve two-stage linear adaptive robust optimization (ARO) problems with binary here-and-now variables and polyhedral uncertainty sets. We encode the optimal here-and-now decisions, the worst-case scenarios associated with the optimal here-and-now decisions, and the optimal wait-and-see decisions into what we denote as the strategy. We solve multiple similar ARO instances in advance using the column and constraint generation algorithm and extract the optimal strategies to generate a training set. We train machine learning models that predict high-quality strategies for the here-and-now decisions, the worst-case scenarios associated with the optimal here-and-now decisions, and the wait-and-see decisions. The models can be applied to problems with varying dimensions. We also introduce novel methods to expedite training data generation and reduce the number of different target classes the machine learning algorithm needs to be trained on. We apply the proposed approach to the facility location, the multi-item inventory control and the unit commitment problems. Our approach solves ARO problems drastically faster than the state-of-the-art algorithms with high accuracy.

Robust combinatorial optimization problems under budgeted interdiction uncertainty

AbstractIn robust combinatorial optimization, we would like to find a solution that performs well under all realizations of an uncertainty set of possible parameter values. How we model this uncertainty set has a decisive influence on the complexity of the corresponding robust problem. For this reason, budgeted uncertainty sets are often studied, as they enable us to decompose the robust problem into easier subproblems. We propose a variant of discrete budgeted uncertainty for cardinality-based constraints or objectives, where a weight vector is applied to the budget constraint. We show that while the adversarial problem can be solved in linear time, the robust problem becomes NP-hard and not approximable. We discuss different possibilities to model the robust problem and show experimentally that despite the hardness result, some models scale relatively well in the problem size.

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.

A robust optimization method for power systems with decision‐dependent uncertainty

AbstractRobust optimization is an essential tool for addressing the uncertainties in power systems. Most existing algorithms, such as Benders decomposition and column‐and‐constraint generation (C&CG), focus on robust optimization with decision‐independent uncertainty (DIU). However, increasingly common decision‐dependent uncertainties (DDUs) in power systems are frequently overlooked. When DDUs are considered, traditional algorithms for robust optimization with DIUs become inapplicable. This is because the previously selected worst‐case scenarios may fall outside the uncertainty set when the first‐stage decision changes, causing traditional algorithms to fail to converge. This study provides a general solution algorithm for robust optimization with DDU, which is called dual C&CG. Its convergence and optimality are proven theoretically. To demonstrate the effectiveness of the dual C&CG algorithm, we used the do‐not‐exceed limit (DNEL) problem as an example. The results show that the proposed algorithm can not only solve the simple DNEL model studied in the literature but also provide a more practical DNEL model considering the correlations among renewable generators.

A data-driven target-oriented robust optimization framework: bridging machine learning and optimization under uncertainty

ABSTRACT The target-oriented robust optimization (TORO) approach converts the original objectives to system targets and instead maximizes an uncertainty budget or robustness index. Machine learning techniques are used to tighten uncertainty sets which also reduce the pessimism of robust solutions. However, existing approaches operate only on a one-way sequential flow, where the data estimation and optimization modules implement their tasks independently. This overlooks the potential of capturing feedback within the solution framework to update forecasts and decisions based on the recent realizations of the uncertainty and system outcomes. This research proposes a novel closed-loop data-driven TORO framework, leveraging on the power of machine learning and feedback systems for optimization under uncertainty. The framework provides an array of solutions for various risk appetites and supports efficient decision-making as computational tractability is retained. A hypothetical case study is solved on a combined inventory and routing problem to demonstrate its applicability and features.

Robust, extended goal programming with uncertainty sets: an application to a multi-objective portfolio selection problem leveraging DEA

AbstractThis study presents a two-phase approach of Data Envelopment Analysis (DEA) and Goal Programming (GP) for portfolio selection, representing a pioneering attempt at combining these techniques within the context of portfolio selection. The approach expands on the conventional risk and return framework by incorporating additional financial factors and addressing data uncertainty, which allows for a thorough examination of portfolio outcomes while accommodating investor preferences and conservatism levels. The initial phase employs a super-efficiency DEA model to streamline asset selection by identifying suitable investment candidates based on efficiency scores, setting the stage for subsequent portfolio optimization. The second phase leverages the Extended GP (EGP) framework, which facilitates the comprehensive incorporation of investor preferences to determine the optimal weights of the efficient assets previously identified within the portfolio. Each goal is tailored to reflect specific financial factors spanning both technical and fundamental aspects. To tackle data uncertainty, robust optimization is applied. The research contributes to the robust GP (RGP) literature by analyzing new RGP variants, overcoming limitations of traditional and other uncertain GP models by incorporating uncertainty sets. Robust counterparts of the EGP models are accordingly developed using polyhedral and combined interval and polyhedral uncertainty sets, providing a flexible representation of uncertainty in financial markets. Empirical results, based on real data from the Tehran Stock Exchange comprising 779 assets, demonstrate the superiority of the proposed approach over traditional portfolio selection methods across various uncertainty settings. Additionally, a comprehensive sensitivity analysis investigates the impact of uncertainty levels on the robust EGP models. The proposed framework offers guidance to investors and fund managers through a pragmatic approach, enabling informed and robust portfolio decisions by considering efficiency, uncertainty, and extended financial factors.

Adjustable light robust optimization with second order stochastic dominance constraints

Robust optimization suffers from excessive conservatism and infeasibility due to stringent requirements on all constraints within the uncertainty set, rendering robust solutions impractical. The light robust approach has been proposed to mitigate these challenges. This method aims to identify a solution that minimizes the weighted sum of all constraint violations while adhering to a predetermined limit on the deterioration of expected return. Drawing inspiration from light robustness, our paper introduces the adjustable light robust optimization models with second-order stochastic dominance constraints. We empirically examine their feasibility, out-of-sample performance, and the dynamic trade-off between return and robustness.

Two‐stage stochastic‐robust planning of distributed energy storage systems with Archimedes optimisation algorithm

AbstractWith the advancement of energy storage technologies, energy storage systems (ESSs) have emerged as a promising solution for distribution networks to mitigate the impact of intermittent and violate renewable energy sources. The optimal planning of distributed ESS is studied to minimise the investment and operational costs for the distribution system operator. To address the various uncertainties associated with load demand and distributed generation, the authors formulate the problem as a two‐stage stochastic‐robust optimisation problem. The proposed formulation implements various representative scenarios of actual operating conditions and constructs the robust uncertainty set to ensure feasibility under worst‐case scenarios. In view of the computational complexity of the proposed model, a solution approach combining the Archimedes optimisation algorithm and the global optimisation method is presented. By decomposing the investment and operation stages, the subproblems are relaxed into mixed integer second‐order cone programming models, which can be solved in parallel based on scenarios. Numerical studies are carried out on a 17‐node test system to demonstrate the validity of the proposed model and algorithm. In addition, a comparison between the proposed method and the genetic algorithm is performed, to illustrate its superiority in solving speed and solution optimality.

Cluster partition-based two-layer expansion planning of grid–resource–storage for distribution networks

In order to realize the optimal planning of grid–resource–storage for distribution networks (DNs) with high penetrated distributed photovoltaics (PVs), a cluster partition-based two-layer expansion planning for DNs is proposed. First, a comprehensive cluster partition index-based cluster partition method is proposed, which involves the indexes such as electrical distance, power balance of the cluster, and cluster size. Second, a cluster partition-based two-layer expansion planning model is proposed. In the upper layer, a line planning model for clusters is established to carry out the planning of cluster connection lines. In the lower layer, a robust source-storage planning model is established with the uncertainty of PVs and loads, and then, the optimal location and capacity of PVs and energy storages (ESs) can be obtained. In addition, the uncertainty regulation parameter is utilized to control the range of uncertainty sets, which can reduce the conservatism of the optimization. Finally, the proposed method is carried out in a real DN in China, which can effectively improve the economy of DN planning.

Adaptive dynamic programming and distributionally robust optimal control of linear stochastic system using the Wasserstein metric

SummaryIn this paper, we consider the optimal control of unknown stochastic dynamical system for both the finite‐horizon and infinite‐horizon cases. The objective of this paper is to find an optimal controller to minimize the expected value of a function which depends on the random disturbance. Throughout this paper, it is assumed that the mean vector and covariance matrix of the disturbance distribution is unknown. An uncertainty set in the space of mean vector and the covariance matrix is introduced. For the finite‐horizon case, we derive a closed‐form expression of the unique optimal policy and the opponents policy that generates the worst‐case distribution. For the infinite‐horizon case, we simplify the Riccati equation obtained in the finite‐hozion setting to an algebraic Riccati equation, which can guarantee the existence of the solution of the Riccati equation. It is shown that the resulting optimal policies obtained in these two cases can stabilize the expected value of the system state under the worst‐case distribution. Furthermore, the unknown system matrices can also be explicitly computed using the adaptive dynamic programming technique, which can help compute the optimal control policy by solving the algebraic Riccati equation. Finally, a simulation example is presented to demonstrate the effectiveness of our theoretical results.

Robust maximum fairness consensus models with aggregation operator based on data-driven method

In group decision-making (GDM), when decision-makers (DMs) feel it is unfair, they may take uncooperative measures to disrupt the consensus-reaching process (CRP). On the other hand, it is difficult for the moderator to objectively determine each DM’s unit consensus cost and weight in CRP. Hence, this paper proposes data-driven robust maximum fairness consensus models (RMFCMs) to address these. First, this paper uses the robust optimization method to construct multiple uncertainty sets to describe the uncertainty of the DMs’ unit adjustment cost and proposes the RMFCMs. Subsequently, based on the DMs’ historical data, the DMs’ weights in the CRP are determined by a data-driven method based on the kernel density estimation (KDE) method. Finally, this paper also applies the proposed models to the carbon emission reduction negotiation process between governments and enterprises, and the experimental results verify the rationality and robustness of the proposed consensus model.

A robust optimization method for new distribution systems based on adaptive data-driven polyhedral sets

In order to better describe the uncertainty of renewable energy output, this paper proposed a novel robust optimization method for new distribution systems based on adaptive data-driven polyhedral sets. First, an ellipsoidal uncertainty set was established using historical data on renewable energy output, and a data-driven convex hull polyhedral set was established by connecting high-dimensional ellipsoidal vertices; on this basis, an adaptive data-driven polyhedral set model was established to address the problem of high conservatism in the scaling process of convex hull polyhedral sets. Furthermore, a novel adaptive data-driven robust scheduling model for new distribution systems was established, and a column-and-constraint generation (C&CG) algorithm was used to solve the robust scheduling model. Finally, the improved IEEE-33 bus system simulation verification shows that the robust scheduling model for new distribution systems based on adaptive data-driven polyhedral sets can reduce conservatism and improve the robustness of optimization results.

Impact of α enhancement on the asteroseismic age determination of field stars

Aims. We investigated the theoretical biases affecting the asteroseismic grid-based estimates of stellar mass, radius, and age in the presence of a mismatch between the heavy element mixture of observed stars and stellar models. Methods. We performed a controlled simulation adopting a stellar effective temperature, [Fe/H], an average large frequency spacing, and a frequency of maximum oscillation power as observational constraints. Synthetic stars were sampled from grids of stellar models computed with different [α/Fe] values from 0.0 to 0.4. The mass, radius, and age of these objects were then estimated by adopting a grid of models with a fixed [α/Fe] value of 0.0. The experiment was repeated assuming different sets of observational uncertainties. In the reference scenario, we adopted an uncertainty of 1.5% in seismic parameters, 50 K in effective temperature, and 0.05 dex in [Fe/H]. A higher uncertainty in the atmospheric constraints was also adopted in order to explore the impact on the precision of the observations of the estimated stellar parameters. Results. Our Monte Carlo experiment showed that estimated parameters are biased up to 3% in mass, 1.5% in radius, and 4% in age when the reference uncertainty scenario was adopted. These values correspond to 45%, 48%, and 16% of the estimated uncertainty in the stellar parameters. These non-negligible biases in mass and radius disappear when adopting larger observational uncertainties because of the possibility of the fitting algorithm exploring a wider range of possible solutions. However, in this scenario, the age is significantly biased by −8%. Finally, we verified that the stellar mass, radius, and age can be estimated with a high accuracy by adopting a grid with the incorrect value of [α/Fe] if the metallicity [Fe/H] of the target is adjusted to match the Z in the fitting grid. In this scenario, the maximum bias in the age was reduced to 1.5%.

Evaluating Adaptation Pathways in a Complex Multi‐Risk System

AbstractDisaster Risk Management (DRM) is complex due to interacting climate risks from interacting hazards and sectors. We develop a synthetic multi‐risk test case to explore the effects of these interactions on decision‐making under deep uncertainty. The test case accounts for changes in hazard impacts and occurrence frequency due to interactions between floods and droughts. Interactions between the shipping, housing and agricultural sectors are considered as well. We use this test case to explore the utility of the Dynamic Adaptive Policy Pathways for Multi‐Risk (DAPP‐MR) framework. DAPP‐MR has been introduced to assess DRM policies' effectiveness under deep uncertainties and to develop integrated adaptive strategies considering interactions across hazards, sectors and time. With the test case, we show that the stage‐wise approach of DAPP‐MR, which gradually increases the complexity of the analysis, can facilitate the evaluation process. Earlier stages of the analysis can be used to establish the direct cause‐effect relations, later stages allow us to identify whether additional interacting factors have a significant effect on the direct cause‐effect relations. As a result, decision‐makers can gain insights into dependencies and their relevance for developing short‐to long‐term strategies under deep uncertainty. We show that multi‐risk interactions can lead to non‐linear effects that influence the outcome of the policy analysis, for example, the indirect influence of a decision in one sector on another. Future work could investigate further improving the operationalization of this staged approach as well as extending the set of uncertainties, dynamics and decision‐rules accounted for in the multi‐risk test case.

A data-driven distributionally robust optimization approach for the core acquisition problem

Reusing electric vehicles (EV) batteries that reach the end of their useful first life is an environmental and cost-competitive option; however, the process of recycling EV batteries is not yet mature. Due to complex electrochemical reactions and physical conditions, the quality of used EV batteries (cores) is highly uncertain. The remanufacturer needs to make the acquisition decision under quality distributional ambiguity. Perfect quality distribution of cores cannot be known to the remanufacturer in practice. We develop distributionally robust optimization models based on phi-divergence measures and the imprecise Dirichlet model (DRO-IDM) to derive robust decisions. First, we find that the bounds of quality probability intervals are identified solely based on the collected data by introducing the imprecise Dirichlet model. The derived finite-sample boundary can reduce the scope of the uncertainty set and avoid the no-direction search issue. Second, our models can hedge against distributional uncertainty, reduce the probability of a robust solution that deviates from the optimal solution, and correct bias in decision making. Third, we extend the DRO-IDM to develop data-driven models, that can reassess the value of multisource quality information to improve the estimation accuracy of core quality and maximize the remanufacturer’s profit. Our study provides new insights for remanufacturers: the new remanufacturing process proposed in our work can assist remanufacturers in utilizing the values of cores without disassembly; the information-aware algorithm can offer the remanufacturing sector a valuable tool for efficiently filtering out invalid information in optimizing acquisition decisions; this capability empowers decision-makers to leverage multiple sources of information and expedite the process of digital transformation in remanufacturing; our approach can also provide a manner of integrating information fusion and distribution learning into remanufacturing.

Multi time scale optimal dispatch of distribution network considering load demand response

Considering the uncertainty of wind power output and load as well as the differences in the operating characteristics of various adjustable resources over different time scales, a multitime scale optimal dispatching model based on stochastic optimization and distributed robust optimization is designed for the distribution network. Firstly, the demand response of the user load side is divided into price-based load and incentive-based load, and price elasticity coefficients are introduced to describe it. Then, with the minimum operation cost of the distribution network, the loss cost, and the penalty cost of abandoned wind as the objective function, various active and reactive regulation elements as well as the system power flow formulas are taken as constraints. In the day-ahead and real-time stages, a stochastic optimization method based on scenarios and opportunity constraints is introduced. In the intraday stage, a self-regressive sliding window prediction model is used to predict wind power and load processing, and a Wasserstein distance-based prediction error uncertainty set is established to establish an intraday distributed robust rolling optimization model. The improved IEEE 33-node case is used for simulation verification, and the results show that the model can effectively reduce system costs and improve system operation stability.