Distributionally Robust Optimization: A review on theory and applications

Operations research (OR) techniques have been widely used for optimizing problems, such as manufacturing scheduling, supply chain optimization, and resource allocation. Despite its effectiveness, traditional OR, especially exact methods, often struggle with scalability, computational efficiency, and adaptability to the dynamic and uncertain environments of Industry 4.0. While machine learning (ML) advancements provide novel approaches for addressing these challenges, they also present limitations, such as the lack of guaranteeing exact solutions and the need of relevant data. Therefore, the integration of OR and ML offers a balanced solution, leveraging ML’s capability to extract patterns from large datasets and making predictive decisions and OR’s precision to enhance decision-making processes, especially in scheduling tasks withing the context of Industry 4.0. This combination not only improves solution robustness and efficiency but also mitigates individual limitations of both fields. and make predictive decisions under uncertainty complements the decision-making process of OR. This paper aims to conduct a bibliometric analysis and a brief literature review on the integration of ML and OR, focusing on their application in scheduling problems.

Distributionally robust optimization (DRO) has been introduced for solving stochastic programs in which the distribution of the random variables is unknown and must be estimated by samples from that distribution. A key element of DRO is the construction of the ambiguity set, which is a set of distributions that contains the true distribution with a high probability. Assuming that the true distribution has a probability density function, we propose a class of ambiguity sets based on confidence bands of the true density function. As examples, we consider the shape-restricted confidence bands and the confidence bands constructed with a kernel density estimation technique. The former allows us to incorporate the prior knowledge of the shape of the underlying density function (e.g., unimodality and monotonicity), and the latter enables us to handle multidimensional cases. Furthermore, we establish the convergence of the optimal value of DRO to that of the underlying stochastic program as the sample size increases. The DRO with our ambiguity set involves functional decision variables and infinitely many constraints. To address this challenge, we apply duality theory to reformulate the DRO to a finite-dimensional stochastic program, which is amenable to a stochastic subgradient scheme as a solution method.

Quay cranes are among the most important resources in port terminals, and their efficient use is crucial for port terminals to remain competitive in the market. The problem of determining the sequence of tasks performed by the quay cranes that minimizes the turnaround time of a vessel is an NP‐hard problem known as the quay crane scheduling problem (QCSP).In this paper, we consider the unidirectional QCSP under uncertain processing times, where the cranes are only allowed to move in a specific direction. We start by presenting the first distributionally robust optimization (DRO) model for this problem and an exact decomposition algorithm to solve it. The proposed DRO model makes it possible to derive a stochastic programming model and a robust optimization model for the unidirectional QCSP by an appropriate choice of the risk‐averse parameter of the model. Through extensive numerical results, we compare these three approaches—stochastic programming, robust optimization, and DRO—to investigate whether significant differences among the solutions obtained exist. Finally, we propose a new method for helping practitioners to determine a representative set of different DRO solutions.

Chapter 4 - Data-driven robust stochastic optimization for power systems operations

This book covers techniques that can be used to analyze data from IoT sensors and addresses questions regarding the performance of an IoT system. It strikes a balance between practice and theory so one can learn how to apply these tools in practice with a good understanding of their inner workings. This is an introductory book for readers who have no familiarity with these techniques. The techniques presented in An Introduction to IoT Analytics come from the areas of machine learning, statistics, and operations research. Machine learning techniques are described that can be used to analyze IoT data generated from sensors for clustering, classification, and regression. The statistical techniques described can be used to carry out regression and forecasting of IoT sensor data and dimensionality reduction of data sets. Operations research is concerned with the performance of an IoT system by constructing a model of the system under study and then carrying out a what-if analysis. The book also describes simulation techniques. Key Features IoT analytics is not just machine learning but also involves other tools, such as forecasting and simulation techniques. Many diagrams and examples are given throughout the book to fully explain the material presented. Each chapter concludes with a project designed to help readers better understand the techniques described. The material in this book has been class tested over several semesters. Practice exercises are included with solutions provided online at www.routledge.com/9780367686314 Harry G. Perros is a Professor of Computer Science at North Carolina State University, an Alumni Distinguished Graduate Professor, and an IEEE Fellow. He has published extensively in the area of performance modeling of computer and communication systems.

We investigate the use of distributionally robust optimization (DRO) as a tractable tool to recover the asymptotic statistical guarantees provided by the Central Limit Theorem, for maintaining the feasibility of an expected value constraint under ambiguous probability distributions. We show that using empirically defined Burg-entropy divergence balls to construct the DRO can attain such guarantees. These balls, however, are not reasoned from the standard data-driven DRO framework since by themselves they can have low or even zero probability of covering the true distribution. Rather, their superior statistical performances are endowed by linking the resulting DRO with empirical likelihood and empirical processes. We show that the sizes of these balls can be optimally calibrated using chi-square process excursion. We conduct numerical experiments to support our theoretical findings.

Problem definition: In this paper, we study the fleet repositioning problem for a free-float vehicle sharing system, aiming to dynamically match the vehicle supply and travel demand at the lowest total cost of repositioning and lost sales. Academic/practical relevance: Besides the analytical results on the optimal repositioning policy, the proposed optimization framework is applicable to practical problems by its computational efficiency as well as the capability to handle temporally dependent demands. Methodology: We first formulate the problem as a stochastic dynamic program. To solve for a multiregion system, we deploy the distributionally robust optimization (DRO) approach that can incorporate demand temporal dependence, motivated by real data. We first propose a “myopic” two-stage DRO model that serves as both an illustration of the DRO framework and a benchmark for the later multistage model. We then develop a computationally efficient multistage DRO model with an enhanced linear decision rule (ELDR). Results: Under a two-region system, we find a simple reposition up-to and down-to policy to be optimal, when the demands are temporally independent. Such a structure is also preserved by our ELDR solution. We also provide new analytical insights by proving the optimality of ELDR in solving the single-period DRO problem. We then show that the numerical performance of the ELDR solution is close to the exact optimal solution from the dynamic program. Managerial implications: In a real-world case study of car2go, we quantify the “value of repositioning” and compare with several benchmarks to demonstrate that the ELDR solutions are computationally scalable and in general result in lower cost with less frequent repositioning. We also explore several managerial implications and extensions from the experiments.

In this golden age of artificial intelligence, transparency and responsible decision-making are paramount. While machine learning (ML) and operational research (OR) optimisations are fundamental aspects of AI, the benefits of explainable AI (XAI) for combinatorial optimisations remain underexplored. This study investigates the convergence of XAI and OR, emphasising the importance of transparency in combinatorial optimisations. Using the Knapsack problem as an example, we demonstrate that interpretable ML models can effectively solve combinatorial optimisation challenges and enhance transparency. Additionally, we illustrate the application of post-hoc XAI methods to OR optimisations solved with ML, providing transparent, human-friendly explanations. The key contributions of this work include proposing the application of the SAGE framework for transparent OR, demonstrating the integration of XAI with combinatorial optimisations, and offering practical guidelines for creating transparent explanations. These contributions can aid decision-makers in understanding, communicating, and trusting combinatorial optimisation solutions, paving the way for enhanced transparency in operational research across various sectors.

Coordinated operations of electricity and district heating networks offer a potential for mitigating inherent variability of renewable energy sources (RES) in the ongoing transition to smart grids. This paper proposes a two-stage distributionally robust optimization (DRO) approach to determine the optimal day-ahead unit commitment in coordinated electricity and district heating networks with variable RES power output. The proposed formulation is to minimize the worst-case expected total cost over an ambiguity set comprising a family of probability distributions with given support and moments of RES power output. As such, the proposed DRO approach can overcome the limitations of stochastic programming in its inherent dependence of exact probability distributions along with a huge computational burden, but also becomes less conservative than classical robust optimization. The pertinent DRO model is eventually reformulated as a tractable mixed-integer second-order cone (SOC) programming after employing linear decision rules and the SOC duality. Simplified affine policies are utilized to further improve computational tractability and performance. Finally, case studies are conducted based on Barry Island electricity and district heating networks. The numerical results demonstrate the decision-making superiority of the proposed method as compared with deterministic, stochastic programming, and robust optimization approaches. They also validate the computational improvement of the proposed approach by employing simplified affine policies.

Distributionally robust facility location with uncertain facility capacity and customer demand

Distributionally robust optimization for collaborative emergency response network design

Clean energy resources, like wind, have a stochastic nature, which involves uncertainties in the power system. Introducing energy storage systems (ESS) to the network can compensate for the uncertainty in wind plant output and allow the plant to participate in ancillary service markets. Advance in compressed air energy storage system (CAES) technologies and their fast response make them suitable for ancillary services. This paper investigates the participation of a combined energy system composed of wind plants and compressed air energy storage system (CAES) in the energy market from a private owner’s viewpoint, including trading in energy markets and bidding for frequency regulation and reserve capacity in ancillary service markets. Since this problem contains various uncertainties associated with market prices, wind generation levels, and regulation signals, distributionally robust optimization (DRO) is used to model the uncertainties and enhance the simultaneous participation of a combined wind-CAES system in day-ahead energy and ancillary service markets. This method combines the advantages of stochastic and robust optimization. In contrast to robust optimization (RO), the method consolidates specific statistical data to reduce conservative results. Simulation results demonstrate the proposed model’s effectiveness in handling uncertainties and provide a framework for investors in this area. In addition, case study analyses are applied to assess the model’s performance and validate the coordination of a wind plant and compressed air energy storage system in participating in a deregulated electricity market. Finally, DRO and RO are compared in modeling the uncertainties of the optimization problem. The optimal outputs demonstrate the effectiveness of DRO in terms of achieving higher realized profits with less conservative results.

Randomized decision making refers to the process of making decisions randomly according to the outcome of an independent randomization device, such as a dice roll or a coin flip. The concept is unconventional, and somehow counterintuitive, in the domain of mathematical programming, in which deterministic decisions are usually sought even when the problem parameters are uncertain. However, it has recently been shown that using a randomized, rather than a deterministic, strategy in nonconvex distributionally robust optimization (DRO) problems can lead to improvements in their objective values. It is still unknown, though, what is the magnitude of improvement that can be attained through randomization or how to numerically find the optimal randomized strategy. In this paper, we study the value of randomization in mixed-integer DRO problems and show that it is bounded by the improvement achievable through its continuous relaxation. Furthermore, we identify conditions under which the bound is tight. We then develop algorithmic procedures, based on column generation, for solving both single- and two-stage linear DRO problems with randomization that can be used with both moment-based and Wasserstein ambiguity sets. Finally, we apply the proposed algorithm to solve three classical discrete DRO problems: the assignment problem, the uncapacitated facility location problem, and the capacitated facility location problem and report numerical results that show the quality of our bounds, the computational efficiency of the proposed solution method, and the magnitude of performance improvement achieved by randomized decisions. Summary of Contribution: In this paper, we present both theoretical results and algorithmic tools to identify optimal randomized strategies for discrete distributionally robust optimization (DRO) problems and evaluate the performance improvements that can be achieved when using them rather than classical deterministic strategies. On the theory side, we provide improvement bounds based on continuous relaxation and identify the conditions under which these bound are tight. On the algorithmic side, we propose a finitely convergent, two-layer, column-generation algorithm that iterates between identifying feasible solutions and finding extreme realizations of the uncertain parameter. The proposed algorithm was implemented to solve distributionally robust stochastic versions of three classical optimization problems and extensive numerical results are reported. The paper extends a previous, purely theoretical work of the first author on the idea of randomized strategies in nonconvex DRO problems by providing useful bounds and algorithms to solve this kind of problems.

Reliable Facility Location Under Event-Correlated Uncertainty

Advanced AI-Generated Content (AIGC) technologies have injected new impetus into teleoperation, enhancing its security and efficiency. Edge AIGC networks have been introduced to meet the stringent low-latency requirements of teleoperation. However, the inherent uncertainty of AIGC service quality and the need to incentivize AIGC service providers (ASPs) make the design of a robust incentive mechanism essential. This design is particularly challenging due to uncertainty and information asymmetry, as teleoperators have limited knowledge of the remaining resource capacities of ASPs. To this end, we propose a distributionally robust optimization (DRO)-based contract theory to design robust reward schemes for AIGC task offloading. Notably, our work extends the contract theory by integrating DRO, addressing the fundamental challenge of contract design under uncertainty. In this paper, we employ contract theory to model information asymmetry while utilizing DRO to capture the uncertainty in AIGC service quality. Given the inherent complexity of the original DRO-based contract theory problem, we reformulate it into an equivalent, tractable bi-level optimization problem. To efficiently solve this problem, we develop a Block Coordinate Descent (BCD)-based algorithm to derive robust reward schemes. Simulation results on our unitybased teleoperation platform demonstrate that the proposed method improves teleoperator utility by 2.7% to 10.74% under varying degrees of AIGC service quality shifts and increases ASP utility by 60.02% compared to the SOTA method, i.e., Deep Reinforcement Learning (DRL)-based contract theory. The code and data are publicly available at https://github.com/Zijun0819/DROContract-Theory