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In this paper we investigate the dynamics of a duopoly game with ambiguity aversion regarding uncertainty in demand and constant expectations concerning competitor production. The focus is on an asymmetric Cournot game where players engage in robust optimization and have different beliefs about the possible realizations of the random parameters of the price function. The players' ambiguity aversion introduces multiple equilibria and instability that otherwise would not be present. The investigation of the global dynamics of the game reveals the emergence, through border-collision bifurcations, of periodic and chaotic dynamics. [ABSTRACT FROM AUTHOR]

This article studies robust optimal asset-liability management problems for an ambiguity-averse manager in a possibly non-Markovian environment with stochastic investment opportunities. The manager has access to one risk-free asset and one risky asset in a financial market. The market price of risk relies on a stochastic factor process satisfying an affine-form, square-root, Markovian model, whereas the risky asset’s return rate and volatility are potentially given by general non-Markovian, unbounded stochastic processes. This financial framework includes, but is not limited to, the constant elasticity of variance (CEV) model, the family of 4/2 stochastic volatility models, and some path-dependent non-Markovian models, as exceptional cases. As opposed to most of the papers using the Hamilton-Jacobi-Bellman-Issacs (HJBI) equation to deal with model ambiguity in the Markovian cases, we address the non-Markovian case by proposing a backward stochastic differential equation (BSDE) approach. By solving the associated BSDEs explicitly, we derive, in closed form, the robust optimal controls and robust optimal value functions for power and exponential utility, respectively. In addition, analytical solutions to some particular cases of our model are provided. Finally, the effects of model ambiguity and market parameters on the robust optimal investment strategies are illustrated under the CEV model and 4/2 model with numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we develop an exact reformulation and a deterministic approximation for distributionally robust joint chance-constrained programmings $ ({\rm DRCCPs}) $ (DRCCPs) with a general class of convex uncertain constraints under data-driven Wasserstein ambiguity sets. It is known that robust chance constraints can be conservatively approximated by worst-case conditional value-at-risk (CVaR) constraints. It is shown that the proposed worst-case CVaR approximation model can be reformulated as an optimization problem involving biconvex constraints for joint DRCCP. This approximation is essentially exact under certain conditions. We derive a convex relaxation of this approximation model by constructing new decision variables which allows us to eliminate biconvex terms. Specifically, when the constraint function is affine in both the decision variable and the uncertainty, the resulting approximation model is equivalent to a tractable mixed-integer convex reformulation for joint binary DRCCP. Numerical results illustrate the computational effectiveness and superiority of the proposed formulations. [ABSTRACT FROM AUTHOR]

Multistage mixed-integer distributionally robust optimization (DRO) forms a class of extremely challenging problems since their size grows exponentially with the number of stages. One way to model the uncertainty in multistage DRO is by creating sets of conditional distributions (the so-called conditional ambiguity sets) on a finite scenario tree and requiring that such distributions remain close to nominal conditional distributions according to some measure of similarity/distance (e.g., Φ-divergences or Wasserstein distance). In this paper, new bounding criteria for this class of difficult decision problems are provided through scenario grouping using the ambiguity sets associated with various commonly used Φ-divergences and the Wasserstein distance. Our approach does not require any special problem structure such as linearity, convexity, stagewise independence, and so forth. Therefore, while we focus on multistage mixed-integer DRO, our bounds can be applied to a wide range of DRO problems including two-stage and multistage, with or without integer variables, convex or nonconvex, and nested or nonnested formulations. Numerical results on a multistage mixed-integer production problem show the efficiency of the proposed approach through different choices of partition strategies, ambiguity sets, and levels of robustness. [ABSTRACT FROM AUTHOR]

We propose an end-to-end distributionally robust system for portfolio construction that integrates the asset return prediction model with a distributionally robust portfolio optimization model. We also show how to learn the risk-tolerance parameter and the degree of robustness directly from data. End-to-end systems have an advantage in that information can be communicated between the prediction and decision layers during training, allowing the parameters to be trained for the final task rather than solely for predictive performance. However, existing end-to-end systems are not able to quantify and correct for the impact of model risk on the decision layer. Our proposed distributionally robust end-to-end portfolio selection system explicitly accounts for the impact of model risk. The decision layer chooses portfolios by solving a minimax problem where the distribution of the asset returns is assumed to belong to an ambiguity set centered around a nominal distribution. Using convex duality, we recast the minimax problem in a form that allows for efficient training of the end-to-end system. [ABSTRACT FROM AUTHOR]

We study adjustable distributionally robust optimization problems, where their ambiguity sets can potentially encompass an infinite number of expectation constraints. Although such ambiguity sets have great modeling flexibility in characterizing uncertain probability distributions, the corresponding adjustable problems remain computationally intractable and challenging. To overcome this issue, we propose a greedy improvement procedure that consists of solving, via the (extended) linear decision rule approximation, a sequence of tractable subproblems—each of which considers a relaxed and finitely constrained ambiguity set that can be iteratively tightened to the infinitely constrained one. Through three numerical studies of adjustable distributionally robust optimization models, we show that our approach can yield improved solutions in a systematic way for both two-stage and multistage problems. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: Financial support by the Early Career Scheme from the Hong Kong Research Grants Council [Project No. CityU 21502820], the CityU Start-Up Grant [Project No. 9610481], the CityU Strategic Research Grant [Project No. 7005688], the National Natural Science Foundation of China [Project No. 72032005], and Chow Sang Sang Group Research Fund sponsored by Chow Sang Sang Holdings International Limited [Project No. 9229076] is gratefully acknowledged. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information (https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2021.0181), as well as from the IJOC GitHub software repository (https://github.com/INFORMSJoC/2021.0181). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/. [ABSTRACT FROM AUTHOR]

We seek to provide practicable approximations of the two-stage robust stochastic optimization model when its ambiguity set is constructed with an f-divergence radius. These models are known to be numerically challenging to various degrees, depending on the choice of the f-divergence function. The numerical challenges are even more pronounced under mixed-integer first-stage decisions. In this paper, we propose novel divergence functions that produce practicable robust counterparts, while maintaining versatility in modeling diverse ambiguity aversions. Our functions yield robust counterparts that have comparable numerical difficulties to their nominal problems. We also propose ways to use our divergences to mimic existing f-divergences without affecting the practicability. We implement our models in a realistic location-allocation model for humanitarian operations in Brazil. Our humanitarian model optimizes an effectiveness-equity trade-off, defined with a new utility function and a Gini mean difference coefficient. With the case study, we showcase (1) the significant improvement in practicability of the robust stochastic optimization counterparts with our proposed divergence functions compared to existing f-divergences, (2) the greater equity of humanitarian response that the objective function enforces and (3) the greater robustness to variations in probability estimations of the resulting plans when ambiguity is considered. [ABSTRACT FROM AUTHOR]

In this paper, we study the stochastic optimization problem with multivariate second-order stochastic dominance (MSSD) constraints where the distribution of uncertain parameters is unknown. Instead, only some historical data are available. Using the Wasserstein metric, we construct an ambiguity set and develop a data-driven distributionally robust optimization model with multivariate second-order stochastic dominance constraints (DROMSSD). By utilizing the linear scalarization function, we transform MSSD constraints into univariate constraints. We present a stability analysis focusing on the impact of the variation of the ambiguity set on the optimal value and optimal solutions. Moreover, we carry out quantitative stability analysis for the DROMSSD problems as the sample size increases. Specially, in the context of the portfolio, we propose a convex lower reformulation of the corresponding DROMSSD models under some mild conditions. Finally, some preliminary numerical test results are reported. We compare the DROMSSD model with the sample average approximation model through out-of-sample performance, certificate and reliability. We also use real stock data to verify the effectiveness of the DROSSM model. [ABSTRACT FROM AUTHOR]

We investigate the robust chance constrained optimization problem (RCCOP), which is a combination of the distributionally robust optimization (DRO) and the chance constraint (CC). The RCCOP plays an important role in modeling uncertain parameters within a decision-making framework. The chance constraint, which is equivalent to a constraint of Value-at-risk (VaR), is approximated by risk measures such as Entropic Value-at-risk (EVaR) or Conditional Value-at-risk (CVaR) due to its difficulty to be evaluated. An excellent approximation requires both tractability and non-conservativeness. In addition, the DRO assumes that we know partial information about the distribution of uncertain parameters instead of their known true underlying probability distribution. In this article, we develop a novel approximation EVaR- PC based on EVaR for CC. Then, we evaluate the proposed approximation EVaR- PC using a discrepancy-based ambiguity set with the wasserstein distance. From a theoretical perspective, the EVaR- PC is less conservative than EVaR and the wasserstein distance possesses many good theoretical properties; from a practical perspective, the discrepancy-based ambiguity set can make full use of the data to estimate the nominal distribution and reduce the sensitivity of decisions to priori knowledges. To show the advantages of our method, we show its application in portfolio management in detail and give the relevant experimental results. [ABSTRACT FROM AUTHOR]

Consider a robust retirement decision problem for a risk- and ambiguity-averse investor concerned about return ambiguity in risky asset prices. When the investor aims to maximize the worst-case scenario of his/her utility derived from consumption and bequest, we propose an optimal G-stopping approach to the robust optimization in a dual space with risk ambiguity. Under the G-expectation framework, we consider a reflected G-BSDE with an upper obstacle, which allows us to formulate a parabolic obstacle problem to the dual optimal stopping problem. From the dynamic result in the dual space, we establish the duality theorem to link between the primal problem with return ambiguity and the dual optimal stopping problem with risk ambiguity. We characterize the robust retirement time using the free boundary and derive the robust consumption and investment for a general class of utility functions. [ABSTRACT FROM AUTHOR]

We introduce a new framework, Bayesian distributionally robust optimization (Bayesian-DRO), for data-driven stochastic optimization where the underlying distribution is unknown. Bayesian-DRO contrasts with most of the existing DRO approaches in the use of Bayesian estimation of the unknown distribution. To make computation of Bayesian updating tractable, Bayesian-DRO first assumes the underlying distribution takes a parametric form with unknown parameter and then computes the posterior distribution of the parameter. To address the model uncertainty brought by the assumed parametric distribution, Bayesian-DRO constructs an ambiguity set of distributions with the assumed parametric distribution as the reference distribution and then optimizes with respect to the worst case in the ambiguity set. We show the consistency of the Bayesian posterior distribution and subsequently the convergence of objective functions and optimal solutions of Bayesian-DRO. Our consistency result of the Bayesian posterior requires simpler assumptions than the classical literature on Bayesian consistency. We also consider several approaches for selecting the ambiguity set size in Bayesian-DRO and compare them numerically. Our numerical experiments demonstrate the out-of-sample performance of Bayesian-DRO in comparison with Kullback-Leibler-based DRO (KL-DRO) and Wasserstein-based empirical DRO as well as risk-neutral Bayesian risk optimization. Our numerical results shed light on how to choose the modeling framework (Bayesian-DRO, KL-DRO, Wasserstein-DRO) for specific problems, but the choice for general problems remains an important and open question. [ABSTRACT FROM AUTHOR]

In this paper, we study a distributionally robust optimization (DRO) problem with affine decision rules. In particular, we construct an ambiguity set based on a new family of Wasserstein metrics, shortfall–Wasserstein metrics, which apply normalized utility-based shortfall risk measures to summarize the transportation cost random variables. In this paper, we demonstrate that the multi-dimensional shortfall–Wasserstein ball can be affinely projected onto a one-dimensional one. A noteworthy result of this reformulation is that our program benefits from finite sample guarantee without a dependence on the dimension of the nominal distribution. This distributionally robust optimization problem also has computational tractability, and we provide a dual formulation and verify the strong duality that enables a direct and concise reformulation of this problem. Our results offer a new DRO framework that can be applied in numerous contexts such as regression and portfolio optimization. [ABSTRACT FROM AUTHOR]

We propose two distributionally robust optimization (DRO) models for a mobile facility (MF) fleet-sizing, routing, and scheduling problem (MFRSP) with time-dependent and random demand as well as methodologies for solving these models. Specifically, given a set of MFs, a planning horizon, and a service region, our models aim to find the number of MFs to use (i.e., fleet size) within the planning horizon and a route and time schedule for each MF in the fleet. The objective is to minimize the fixed cost of establishing the MF fleet plus a risk measure (expectation or mean conditional value at risk) of the operational cost over all demand distributions defined by an ambiguity set. In the first model, we use an ambiguity set based on the demand's mean, support, and mean absolute deviation. In the second model, we use an ambiguity set that incorporates all distributions within a 1-Wasserstein distance from a reference distribution. To solve the proposed DRO models, we propose a decomposition-based algorithm. In addition, we derive valid lower bound inequalities that efficiently strengthen the master problem in the decomposition algorithm, thus improving convergence. We also derive two families of symmetry-breaking constraints that improve the solvability of the proposed models. Finally, we present extensive computational experiments comparing the operational and computational performance of the proposed models and a stochastic programming model, demonstrating when significant performance improvements could be gained, and derive insights into the MFRSP. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2022.1153. [ABSTRACT FROM AUTHOR]

Polyhedral coherent risk measures and their worst-case constructions with respect to the ambiguity set are considered. For the case of the discrete distribution and polyhedral ambiguity set, calculating such risk measures reduces to linear programming problems. The distributionally robust portfolio optimization problems based on the reward-risk ratio using worst-case constructions with respect to the polyhedral ambiguity set for these risk measures and average return are analyzed. They are reduced to the appropriate linear programming problems. [ABSTRACT FROM AUTHOR]

We propose a distributionally robust formulation of the traditional risk parity portfolio optimization problem. Distributional robustness is introduced by targeting the discrete probabilities attached to each observation used during parameter estimation. Instead of assuming that all observations are equally likely, we consider an ambiguity set that provides us with the flexibility to find the most adversarial probability distribution based on the investor's desired degree of robustness. This allows us to derive robust estimates to parametrize the distribution of asset returns without having to impose any particular structure on the data. The resulting distributionally robust optimization problem is a constrained convex–concave minimax problem. Our approach is financially meaningful and attempts to attain full risk diversification with respect to the worst-case instance of the portfolio risk measure. We propose a novel algorithmic approach to solve this minimax problem, which blends projected gradient ascent with sequential convex programming. This algorithm is highly flexible and allows the user to choose among alternative statistical distance measures to define the ambiguity set. Moreover, the algorithm is highly tractable and scalable. Our numerical experiments suggest that a distributionally robust risk parity portfolio can yield a higher risk-adjusted rate of return when compared against the nominal portfolio. [ABSTRACT FROM AUTHOR]

We consider a distributionally robust formulation of stochastic optimization problems arising in statistical learning, where robustness is with respect to ambiguity in the underlying data distribution. Our formulation builds on risk-averse optimization techniques and the theory of coherent risk measures. It uses mean–semideviation risk for quantifying uncertainty, allowing us to compute solutions that are robust against perturbations in the population data distribution. We consider a broad class of generalized differentiable loss functions that can be non-convex and non-smooth, involving upward and downward cusps, and we develop an efficient stochastic subgradient method for distributionally robust problems with such functions. We prove that it converges to a point satisfying the optimality conditions. To our knowledge, this is the first method with rigorous convergence guarantees in the context of generalized differentiable non-convex and non-smooth distributionally robust stochastic optimization. Our method allows for the control of the desired level of robustness with little extra computational cost compared to population risk minimization with stochastic gradient methods. We also illustrate the performance of our algorithm on real datasets arising in convex and non-convex supervised learning problems. [ABSTRACT FROM AUTHOR]

Conventional chance-constrained programming methods suffer from the inexactness of the estimated probability distribution of the underlying uncertainty from data. To this end, a distributionally robust approach to the problem allows for a level of ambiguity considered around a reference distribution. In this work, we propose a novel formulation for the distributionally robust chance-constrained programming problem using an ambiguity set constructed from a variant of optimal transport distance that was developed for Gaussian Mixture Models. We show that for multimodal process uncertainty, our proposed method provides an effective way to incorporate statistical moment information into the ambiguity set construction step, thus leading to improved optimal solutions. We illustrate the performance of our method on a numerical example as well as a chemical process case study. We show that our proposed methodology leverages the multimodal characteristics from the uncertainty data to give superior performance over the traditional Wasserstein distance-based method. • Ambiguity set constructed from optimal transport between Gaussian Mixture Models. • Hedging against the right family of candidate distributions to avoid unnecessary conservatism. • A tractable distributionally robust chance constrained optimization formulation. • Applicability to more generate type of uncertain constraints. • Better objective-constraint satisfaction trade-off performance than classical Wasserstein DRCCP model. [ABSTRACT FROM AUTHOR]

We study multistage distributionally robust optimization (DRO) to hedge against ambiguity in quantifying the underlying uncertainty of a problem. Recognizing that not all the realizations and scenario paths might have an "effect" on the optimal value, we investigate the question of how to define and identify critical scenarios for nested multistage DRO problems. Our analysis extends the work of Rahimian, Bayraksan, and Homem-de-Mello [Math. Program., 173 (2019), pp. 393-430], which was in the context of a static/two-stage setting, to the multistage setting. To this end, we define the notions of effectiveness of scenario paths and the conditional effectiveness of realizations along a scenario path for a general class of multistage DRO problems. We then propose easy-to-check conditions to identify the effectiveness of scenario paths in the multistage setting when the distributional ambiguity is modeled via the total variation distance. Numerical results show that these notions provide useful insight on the underlying uncertainty of the problem. [ABSTRACT FROM AUTHOR]

In this paper, we present a sequential sampling-based algorithm for the two-stage distributionally robust linear programming (2-DRLP) models. The 2-DRLP models are defined over a general class of ambiguity sets with discrete or continuous probability distributions. The algorithm is a distributionally robust version of the well-known stochastic decomposition algorithm of Higle and Sen [Math. Oper. Res., 16 (1991), pp. 650-669] for a two-stage stochastic linear program. We refer to the algorithm as the distributionally robust stochastic decomposition (DRSD) method. The key features of the algorithm include (1) it works with data-driven approximations of ambiguity sets that are constructed using samples of increasing size and (2) efficient construction of approximations of the worst-case expectation function that solves only two second-stage subproblems in every iteration. We identify conditions under which the ambiguity set approximations converge to the true ambiguity sets and show that the DRSD method asymptotically identifies an optimal solution, with probability one. We also computationally evaluate the performance of the DRSD method for solving distributionally robust versions of instances considered in stochastic programming literature. The numerical results corroborate the analytical behavior of the DRSD method and illustrate the computational advantage over an external sampling-based decomposition approach (distributionally robust L-shaped method). [ABSTRACT FROM AUTHOR]

Distributionally robust optimization is a popular modeling paradigm in which the underlying distribution of the random parameters in a stochastic optimization model is unknown. Therefore, hedging against a range of distributions, properly characterized in an ambiguity set, is of interest. We study two-stage stochastic programs with linear recourse in the context of distributional ambiguity, and formulate several distributionally robust models that vary in how the ambiguity set is built. We focus on the Wasserstein distance under a p-norm, and an extension, an optimal quadratic transport distance, as mechanisms to construct the set of probability distributions, allowing the support of the random variables to be a continuous space. We study both unbounded and bounded support sets, and provide guidance regarding which models are meaningful in the sense of yielding robust first-stage decisions. We develop cutting-plane algorithms to solve two classes of problems, and test them on a supply-allocation problem. Our numerical experiments provide further evidence as to what type of problems benefit the most from a distributionally robust solution. [ABSTRACT FROM AUTHOR]

This paper studies the fundamental vessel deployment problem in the liner shipping industry, which decides the numbers of mixed-type ships and their sailing frequencies on fixed routes to provide sufficient vessel capacity for fulfilling stochastic shipping demands with high probability. In reality, it is usually difficult (if not impossible) to acquire a precise joint distribution of shipping demands, as they may fluctuate heavily due to the fast-changing economic environment or unpredictable events. To address this challenge, we leverage recent advances in distributionally robust optimization and propose distribution-free robust joint chance constrained models. In the first model, we only assume support, mean as well as lower-order dispersion information of the shipping demands and provide high-quality solutions via a sequential convex optimization algorithm. Comparing with existing literature that chiefly studies individual chance constraints based on concentration inequalities and the union bound, our approach yields solutions that are less conservative and less vulnerable to the magnitude of demand dispersion. We also extend to a data-driven model based on the Wasserstein distance, which suits well in situations where limited historical demand samples are available. Our distributionally robust chance constrained models could serve as a baseline model for vessel deployment, into which we believe additional practical constraints could be incorporated seamlessly. • Distributionally robust joint chance constrained models for the vessel deployment problem. • Examples on the meaning and applications of the mean and dispersion ambiguity set in maritime industry. • Extensive experiments in data-driven setting. • More robust but less conservative deployment plans. [ABSTRACT FROM AUTHOR]

Choice of a risk measure for quantifying risk of an investment portfolio depends on the decision maker's risk preference. In this paper, we consider the case when such a preference can be described by a law invariant coherent risk measure but the choice of a specific risk measure is ambiguous. We propose a robust spectral risk approach to address such ambiguity. Differing from Wang and Xu (SIAM J Optim 30(4):3198–3229, 2020), the new robust model allows one to elicit the decision maker's risk preference through pairwise comparisons and use the elicited preference information to construct an ambiguity set of risk spectra. The robust spectral risk measure (RSRM) is based on the worst case risk spectrum from the set. To calculate RSRM and solve the associated optimal decision making problem, we use a technique from Acerbi and Simonetti (Portfolio optimization with spectral measures of risk. Working paper, 2002) to develop a new computational approach which is independent of order statistics and reformulate the robust spectral risk optimization problem as a single deterministic convex programming problem when the risk spectra in the ambiguity set are step-like. Moreover, we propose an approximation scheme when the risk spectra are not step-like and derive a bound for the model approximation error and its propagation to the optimal decision making problems. Some preliminary numerical test results are reported about the performance of the robust model and the computational scheme. [ABSTRACT FROM AUTHOR]

Distributionally robust optimization (DRO) is a modeling framework in decision making under uncertainty in which the probability distribution of a random parameter is unknown although its partial information (e.g., statistical properties) is available. In this framework, the unknown probability distribution is assumed to lie in an ambiguity set consisting of all distributions that are compatible with the available partial information. Although DRO bridges the gap between stochastic programming and robust optimization, one of its limitations is that its models for large-scale problems can be significantly difficult to solve, especially when the uncertainty is of high dimension. In this paper, we propose computationally efficient inner and outer approximations for DRO problems under a piecewise linear objective function and with a moment-based ambiguity set and a combined ambiguity set including Wasserstein distance and moment information. In these approximations, we split a random vector into smaller pieces, leading to smaller matrix constraints. In addition, we use principal component analysis to shrink uncertainty space dimensionality. We quantify the quality of the developed approximations by deriving theoretical bounds on their optimality gap. We display the practical applicability of the proposed approximations in a production–transportation problem and a multiproduct newsvendor problem. The results demonstrate that these approximations dramatically reduce the computational time while maintaining high solution quality. The approximations also help construct an interval that is tight for most cases and includes the (unknown) optimal value for a large-scale DRO problem, which usually cannot be solved to optimality (or even feasibility in most cases). Summary of Contribution: This paper studies an important type of optimization problem, that is, distributionally robust optimization problems, by developing computationally efficient inner and outer approximations via operations research tools. Specifically, we consider several variants of such problems that are practically important and that admit tractable yet large-scale reformulation. We accordingly utilize random vector partition and principal component analysis to derive efficient approximations with smaller sizes, which, more importantly, provide a theoretical performance guarantee with respect to low optimality gaps. We verify the significant efficiency (i.e., reducing computational time while maintaining high solution quality) of our proposed approximations in solving both production–transportation and multiproduct newsvendor problems via extensive computing experiments. [ABSTRACT FROM AUTHOR]

We introduce a new class of distributionally robust optimization problems under decision-dependent ambiguity sets. In particular, as our ambiguity sets, we consider balls centered on a decision-dependent probability distribution. The balls are based on a class of earth mover's distances that includes both the total variation distance and the Wasserstein metrics. We discuss the main computational challenges in solving the problems of interest and provide an overview of various settings leading to tractable formulations. Some of the arising side results, such as the mathematical programming expressions for robustified risk measures in a discrete space, are also of independent interest. Finally, we rely on state-of-the-art modeling techniques from machine scheduling and humanitarian logistics to arrive at potentially practical applications, and present a numerical study for a novel risk-averse scheduling problem with controllable processing times. Summary of Contribution: In this study, we introduce a new class of optimization problems that simultaneously address distributional and decision-dependent uncertainty. We present a unified modeling framework along with a discussion on possible ways to specify the key model components, and discuss the main computational challenges in solving the complex problems of interest. Special care has been devoted to identifying the settings and problem classes where these challenges can be mitigated. In particular, we provide model reformulation results, including mathematical programming expressions for robustified risk measures, and describe how these results can be utilized to obtain tractable formulations for specific applied problems from the fields of humanitarian logistics and machine scheduling. Toward demonstrating the value of the modeling approach and investigating the performance of the proposed mixed-integer linear programming formulations, we conduct a computational study on a novel risk-averse machine scheduling problem with controllable processing times. We derive insights regarding the decision-making impact of our modeling approach and key parameter choices. [ABSTRACT FROM AUTHOR]

Capacity sharing, as a collaborative strategy among manufacturers, aims to alleviate this problem by sharing manufacturing capabilities to meet demand with large fluctuating ranges. This paper explores a novel supply chain network design problem considering capacity sharing with third-party manufacturers. Specifically, the problem involves location choices for owned manufacturing plants and distribution centers and the selection of third-party manufacturers. Third-party manufactured products can only be shipped to distribution centers, whereas products from plants can be shipped directly or through distribution centers to customer zones. In this network, customer demand and the product prices of third-party manufacturers are assumed to be uncertain. To address uncertainties, we formulate a distributionally robust chance-constrained model for the problem. The probability distributions of the uncertainties are characterized using Wasserstein ambiguity sets. The incorporation of chance constraints serves to enhance the level of satisfactory customer demand. To ensure tractability, we reformulate the distributionally robust model as a solvable model employing duality theory. Finally, we conduct numerical experiments based on a real-life manufacturing company to evaluate the effectiveness of the proposed model. The results confirm that the proposed model reduces the cost standard deviation by an average of 21.9% and increases reliability. Our study can present a reliable framework for designing a capacity-sharing supply chain network for manufacturing enterprises that optimizes cost while improving customer satisfaction by over 95%. [ABSTRACT FROM AUTHOR]

This paper studies the resource-constrained project scheduling problem where uncertain durations are caused by various disruptions with unknown probabilistic functions. To achieve a balance between risk control and computational efficiency, we employ distributionally robust optimization minimizing the worst-case Conditional Value-at-Risk when the upper bounds of durations are not explicitly stipulated. We build a path-based two-stage model over a moment-based ambiguity set. The first stage explicitly generates paths satisfying precedence and resource constraints without any information of uncertainties, while the second stage determines the critical paths over the ambiguity set. Thus, we employ an enhanced Benders decomposition to solve this model where a weighted Benders cut may discriminatingly remove arcs on the critical paths. To ease the computation burden, we develop priority constraints for resource flow and a path elimination method to handle the paths generated by the first stage. A numerical experiment shows that this Benders cut together with the path elimination method can solve more complicated instances efficiently. The result further indicates that this model performs better than the classical model of robust optimization when providing upper bounds for durations. We extract some managerial insights that suggest important guidelines for controlling risk in project scheduling. [ABSTRACT FROM AUTHOR]

Distributionally robust optimization has garnered significant attention for its effectiveness in decision-making under uncertainty. However, employing this strategy faces hurdles posed by intractable models and the difficulty in parameter determination while tackling production scheduling issues under uncertainty. This work presents a novel data-driven distributionally robust optimization framework to address these challenges. A data-driven combined ambiguity set, which incorporates Wasserstein distance and moment information, is devised to yield less conservative solutions. Additionally, a data-driven support set established based on an improved kernel technique is introduced to help identify and exclude potential outliers. The relevant ambiguous parameters are determined through bi-level cross-validation. Subsequently, the data-driven distributionally robust optimization model under combined ambiguity is reformulated into tractable by dual theory. The application to industrial scheduling shows that the proposed method can effectively utilize data information and better hedge against uncertainties while obtaining higher profits. • A novel data-driven CDRO-SS framework is proposed. • Uncertainty is characterized by moment information, Wasserstein metric, and improved kernel method. • The robust counterpart formulations are obtained by duality technique. • The out-of-sample performance of different algorithms is analyzed. [ABSTRACT FROM AUTHOR]

In this paper, we consider an optimal robust reinsurance problem in a diffusion model for an ambiguity-averse insurer, who worries about ambiguity and aims to minimize the robust value involving the probability of absolute ruin and a penalization of model ambiguity. It is assumed that the insurer is allowed to purchase per-claim reinsurance to transfer its risk exposure, and that the reinsurance premium is computed according to the mean-variance premium principle which is a combination of the expected-value and variance premium principles. The optimal reinsurance strategy and the associated value function are derived explicitly by applying stochastic dynamic programming and by solving the corresponding boundary-value problem. We prove that there exists a unique point of inflection which relies on the penalty parameter greatly such that the robust value function is strictly concave up to the unique point of inflection and is strictly convex afterwards. It is also interesting to observe that the expression of the optimal robust reinsurance strategy is independent of the penalty parameter and coincides with the one in the benchmark case without ambiguity. Finally, some numerical examples are presented to illustrate the effect of ambiguity aversion on our optimal results. [ABSTRACT FROM AUTHOR]

The fault detection (FD) system design aims at optimizing the trade‐off between false alarm rate (FAR) and fault detection rate (FDR) under stochastic disturbances or uncertainties. A challenging difficulty in practice is the inexact information of stochastic disturbance distribution, that is, the actual distribution deviates from the one used in the design. To address this challenge, a distributionally robust optimization (DRO) approach that accounts for the inexact distribution information is proposed for the parity relation based FD of stochastic discrete‐time linear systems. It introduces moment‐based and entropy‐based ambiguity sets to describe the inexact disturbance distribution. Over such ambiguity sets, the FD system design for a scalar residual maximizes the worst‐case FDR with respect to a reference fault mode, while ensuring a predefined worst‐case FAR. To address the limitation of a scalar residual, the FD test of a vector residual is constructed with respect to a parameterized set of multiple fault modes. The resulting FD tests can be expressed in the same structure as the celebrated generalized likelihood ratio test (GLRT), while only the detection threshold is adjusted to compensate for distribution ambiguity. Moreover, the worst‐case FDR in the presence of any given fault is evaluated by solving another DRO problem. Using a continuous stirred tank reactor example with inexact distribution information, it is demonstrated that the proposed designs achieve desirable performance trade‐off and provide effective worst‐case FDR evaluations, while the GLRT fails to ensure the predefined FAR. [ABSTRACT FROM AUTHOR]

Considering the dynamic and stochasticity of demand for emergency medical service, this paper proposes two multi-period distributionally robust optimization models with first-order moment and Wasserstein ambiguity sets. To handle non-independent and non-identically distributed demand, we construct two different multi-period models and reformulate the two models into mixed-integer second-order cone programming (MISOCP) based on first-order moment and Wasserstein ambiguity sets. Taking into account the problem size increase caused by multiple periods, we develop a lifted polyhedral approximation algorithm to handle large-scale MISOCP. The numerical experiments demonstrate that our algorithm can significantly improve the solution efficiency compared to benchmarks including the outer approximation algorithm and Gurobi solver. Finally, based on real-world data from Montgomery County, Pennsylvania, we perform sensitivity analysis and compare different models. The results indicate that by comprehensively accounting for the dynamic and stochasticity of demand, managers can significantly mitigate cost while maintaining a heightened reliability level. • Emergency medical service system location with joint chance constraints. • Joint chance constraints with Wasserstein ambiguity sets. • Inter-period independent and inter-period correlated multi-period EMS model. • An efficient lifted polyhedral approximation algorithm based on second-order cone polyhedron approximation. [ABSTRACT FROM AUTHOR]

• This paper investigates the decision-dependence relationship between demand response commitment and the corresponding scheduling uncertainty. • A novel data-driven risk-averse tri-level (i.e., outer, middle, inner) two-stage (dispatching & operational) economic dispatch framework is proposed to address the demand response uncertainty. • A decomposition framework embedded with Benders' and Column-and-Constraint generation methods is used to obtain the optimal solution of the proposed distributionally robust optimization model. Demand response (DR) is usually regarded as a valuable balancing and reserve resource that contributes to maintaining the power balance. However, electricity customers can freely decide whether to reduce their electricity consumption or not in the liberalized day-ahead market and therefore DR is difficult to predict. Considering that, this paper investigates a novel tri-level two-stage data-driven / distributionally robust optimization risk-averse and decision-dependence economic dispatch framework to incorporate DR uncertainties into the day-ahead electricity market clearing process. First, DR commitment is made after establishing the decision-dependence relationship between DR commitment and the corresponding dispatching uncertainty. Then, we construct an ambiguity set for the unknown distribution of the DR uncertainty by purely learning from the historical data. Considering the worst-case distribution within the ambiguity set, an optimal strategy is investigated for scheduling the hourly uncertain DR in day-ahead markets. Finally, a decomposition framework embedded with Benders' and Column-and-Constraint generation (CC&G) methods is built for identifying the optimal solution. The effectiveness of the proposed method is investigated through case studies on the IEEE 30 and IEEE 118 test systems. [ABSTRACT FROM AUTHOR]

The increasing penetration of intermittent renewable generation in power systems allows for more available historical data at hand. However, it is challenging for traditional distributionally robust optimization to capture the difference and heterogeneity in historical samples which makes the solution remain highly conservative. To comprehensively characterize the underlying factors of uncertainty, we proposed a data-driven stochastic distributionally robust optimization model for unit commitment via group-wise ambiguity set constructed by deep representation clustering method. The expectation of the worst-case distribution under a group of scenarios is calculated in the model, with assuming that the scenarios belong to different groups which are ambiguity. A tractable approximation of the model is derived to avoid computational burden and we analyze the optimality conditions of the approximate formulation. Simulations on a modified IEEE-118 illustrate that the proposed approach can make a trade-off between stochastic optimization and distributionally robust optimization and has benefits in reducing the operation costs with the capability to hedge against the perturbation of unrelated samples. • SDRUC model with wasserstein metric achieves balance between SO and DRO. • Deep representation clustering captures multi-scale correlations for partitioning. • Efficient formulas handle massive samples with analysis of optimality conditions. [ABSTRACT FROM AUTHOR]

There are numerous industrial settings in which a decision maker must decide whether to enter into long-term contracts to guarantee price (and hence cash flow) stability or to participate in more volatile spot markets. In this paper, we investigate a data-driven distributionally robust optimization (DRO) approach aimed at balancing this tradeoff. Unlike traditional risk-neutral stochastic optimization models that assume the underlying probability distribution generating the data is known, DRO models assume the distribution belongs to a family of possible distributions, thus providing a degree of immunization against unseen and potential worst-case outcomes. We compare and contrast the performance of a risk-neutral model, conditional value-at-risk formulation, and a Wasserstein distributionally robust model to demonstrate the potential benefits of a DRO approach for an "elasticity-aware" price-taking decision maker. • Investigate tradeoffs between long-term supply contracts vs. spot market sales. • Data-driven DRO approach exploiting Wasserstein ambiguity sets. • Compare risk-neutral, conditional value-at-risk, and data-driven DRO approaches. • Case studies in power portfolio optimization for PJM electricity market. [ABSTRACT FROM AUTHOR]

Designing an efficient shelter location planning is crucial to the rapid implementation of relief operations under the uncertain number of casualties. In practice, the uncertain number of casualties is closely related to disaster severity, whereas previous studies ignore such correlations when modeling uncertainties. In this paper, the scenario-wise ambiguity set is adopted to capture the correlation between the uncertain number of casualties and uncertain covariates, i.e., disaster severity. We develop a two-stage scenario-wise distributionally robust (SDR) model, where the shelter location and capacity allocation decisions are made here-and-now, and recourse decisions to transport casualties are made after the uncertainties on the number of casualties and covariate information have been realized. We approximate the non-convex model into a tractable form, i.e., second-order cone programming (SOCP), which can be solved efficiently by an outer approximation (OA) algorithm for a large-scale computation case. The numerical results with real-world data show that covariate integration (CVI) can contribute to saving costs and improving relief efficiency, and illustrate the computational superiority of the proposed OA algorithm. The results further demonstrate that establishing shelters with large capacities and near the affected areas has indeed had a positive impact on improving relief efficiency. • We integrate uncertain covariate information into shelter location modeling. • The scenario-wise ambiguity set is adopted to capture the correlation between uncertain number of casualties and uncertain covariates. • An outer approximation algorithm is developed to solve the large-scale computation cases efficiently. • The real-world case study show that the covariate integration can contribute to saving costs and improving the relief efficiency. [ABSTRACT FROM AUTHOR]

[Display omitted] • A data-driven TSDRO model is developed for refinery planning under uncertainty. • The uncertainty set is constructed by the Wasserstein metric and RKDE approach. • The robust counterpart of the proposed model is reformulated by duality theory. • A real-world case study of an industrial refinery is presented. • The out-of-sample performance of different methods are analyzed. This work investigates the refinery planning problem under uncertainty in product prices. A novel data-driven Wasserstein distributionally robust optimization framework is proposed for handling uncertainties in the refinery-wide planning operations. A data-driven ambiguity set is constructed based on the Wasserstein metric to model the distributional uncertainty. The robust kernel density estimation technique is adopted to establish the support set to reduce the effect of the potential outliers. Based on the derived support set and ambiguity set, a data-driven two-stage distributionally robust optimization model for refinery planning is developed. Then, the robust counterpart of the proposed model is formulated to make the problem computationally tractable. Finally, a real-world case study on a petroleum refinery is presented to illustrate the effectiveness and applicability of the proposed framework. [ABSTRACT FROM AUTHOR]

• A new crude oil scheduling optimization model with a distributionally robust joint chance constraint is proposed to handle multiple uncertainties. • A data-driven ambiguity set based on the Wasserstein distance is formulated to characterize uncertainties. • The DRJCC model is transformed into an MINLP problem by introducing the CVaR, a big-M coefficient, and additional binary variables. • The DRJCC model is compared with traditional optimization models in case studies. Crude oil scheduling optimization is crucial for decreasing the production cost of refineries. However, the feasibility of the optimized schemes is challenged by uncertainties such as possible ship arrival delays and fluctuating crude demands. This study develops a novel data-driven continuous-time optimization model with a distributionally robust joint chance constraint to ensure the overall feasibility probability of the crude oil processing plan under multiple uncertainties. Industrial data are collected to build an ambiguity set using Wasserstein distance to include the potential joint probability distribution of uncertainties. The radius of the ambiguity set is chosen by cross-validation. The model is constructed considering the worst case in the ambiguity set. First, it is formulated as a conditional value-at-risk constrained optimization model. A big-M coefficient and additional binary variables are then utilized to convert the proposed model into a resolvable problem. The efficacy and reliability of the method are explored through case studies. [ABSTRACT FROM AUTHOR]

Peer-to-peer (P2P) energy trading provides potential economic benefits to prosumers. The prosumers are responsible for managing their own resources/reserves within the energy community, especially for photovoltaic (PV). However, the intermittency of PV leaves a major issue for the optimal operation of P2P energy trading. This paper proposes a fully data-driven distributionally robust optimization (DRO) for P2P energy trading. Specifically, both the optimization approach and the ambiguity set of DRO are formed in a data-driven fashion. The proposed formulation minimizes the expected operation cost of each prosumer, which is modeled as a DRO problem considering the operational constraints. A decentralized energy negotiation mechanism and market clearing algorithm are proposed for P2P energy trading based on the alternating direction multiplier method. Furthermore, the ambiguity set is formed by deep Gaussian process under the framework of bootstrap aggregating. Finally, the equivalent linear programming reformulations of the proposed DRO model are carried out and solved in a distributed manner. Numerical results demonstrate that the proposed DRO-based approach has superior performance for handling the randomness of PV generation compared with robust optimization, stochastic programming, and other DRO variants. [Display omitted] • The uncertainties of PV generation are quantified by implicit posterior variational inference deep Gaussian process. • A novel data-driven ambiguity set is proposed for modeling P2P energy trading. • The tailor-made equivalent model is derived based on the proposed ambiguity set. • This is the first attempt to integrate the DRO model and machine learning approach to P2P energy trading. [ABSTRACT FROM AUTHOR]

Demand-side resources (DSR) have been considered as flexible frequency reserve providers in power systems. Given their uncertain nature in the reserve deployment and the load recovery periods, it is necessary to consider these uncertainties in the frequency dynamic constrained unit commitment (FDUC) model. This paper proposes a two-stage distributionally robust FDUC model to minimize the worst-case cost expectation under a set of probability distributions of DSR uncertainties, which are restricted by the supports and moments generated from statistical information. In addition, the cumulative frequency deviation during the entire regulation period is exploited as a frequency dynamic criterion. Particularly, it can quantify the regulation efficacy of the reserve resources with diverse frequency response characteristics, such as the dead-band time and the reserve delivery time. Simulations based on the IEEE 118-bus and 300-bus systems validate the effectiveness of the proposed method in hedging against the DSR uncertainties. • A frequency dynamic unit commitment model considering uncertain demand-side resources (DSR). • The uncertainty of demand-side resources is captured by a moment-based ambiguity set. • A criterion reflecting the efficacy of different frequency regulation resources. • The load recovery property of demand-side resources diminishes their fast response advantage. • below is the edited version: • A frequency dynamic unit commitment model with uncertain demand-side resources. • The uncertainty of demand-side resources is captured by a moment-based ambiguity set. • A criterion reflecting the efficacy of different frequency regulation resources. • Load recovery diminishes the fast response advantage of demand-side resources. [ABSTRACT FROM AUTHOR]

Disaster relief logistics (DRL) provides adequate relief supplies to victims of natural disasters (e.g., earthquakes and volcanic eruptions). This study explicitly considers supplier selection and inventory pre-positioning corresponding to static preparedness decisions, and post-disaster procurement and delivery associated with dynamic response decisions in actual DRL operations. To tackle issues triggered by shortage and surplus of multi-class relief resources, a flexible option contract is adopted to purchase relief items from suppliers. To measure the risk of demand ambiguity, a worst-case mean-quantile-deviation criterion is introduced to reflect the decision-maker's risk-averse attitude. To handle the ambiguity in the probability distribution of demand, a novel two-stage distributionally robust optimization (DRO) model is developed for the addressed DRL problem. The proposed DRO model can be transformed into equivalent mixed-integer linear programs when the ambiguity sets incorporate all distributions within L 1 -norm and joint L 1 - and L ∞ -norms from a nominal (reference) distribution. A computational study of earthquakes in Iran is conducted to illustrate the applicability of the proposed DRO model to real-world problems. The experimental results demonstrate that our proposed DRO model has superior out-of-sample performance and can mitigate the effect of Optimization Bias compared to the traditional stochastic programming model. Some managerial insights regarding the proposed approach are provided based on numerical results. • A distributionally robust DRLP under option contract and demand ambiguity is introduced. • A two-stage mean-quantile-deviation-based DRO model is formulated. • Two kinds of discrepancy-based ambiguity sets are constructed to address the distributional ambiguity and derive the tractable forms. • Extensive numerical experiments are conducted on a realistic case study. [ABSTRACT FROM AUTHOR]

This paper focuses on an emergency rescue network design problem in response to disasters under uncertainty. Considering the limited distribution information of the uncertain demands extracted from the historical data, we use the mean absolute deviation (MAD) that can derive tractable reformulations and better capture outliers and small deviations, to construct a MAD-based ambiguity set. A distributionally robust optimization model is proposed with the objective of minimizing the preparedness cost and the expected penalty cost of demand shortage under the worst-case distribution over the ambiguity set. We analyze the constructed model and provide some features such as the theoretical bounds of the objective value. For large-scale cases, we reformulate the knotty model using the linear decision rule to obtain tight and tractable problems. Computational experiments verify that the out-of-sample performance of the proposed model is better than that of the stochastic optimization model, especially for extreme cases. The MAD-based ambiguity set combined with the approximation technique can reduce the solution time and obtain high-quality solutions. Moreover, the results show that the amount of data has a significant effect on model performance. These results provide references for decision-makers in the practice of emergency response network design. • Emergency response network distributionally robust optimization. • Mean absolute deviation-based ambiguity set. • Reformulation based on the linear decision rule. [ABSTRACT FROM AUTHOR]

A kernel distributionally robust chance-constrained optimization (DRCCP) method is proposed in this study based on the kernel ambiguity set. The kernel ambiguity set is established via the kernel mean embedding (KME) and the maximum mean discrepancy (MMD) between distributions. The proposed approach can be formulated as two different models. The first one is a mixed-integer model employing the indicator function for handling the joint chance constraint. The second one is a continuous optimization model using the Conditional Value-at-Risk (CVaR) approximation to approximate the indicator function. The proposed method is compared with the popular Wasserstein ambiguity set based approach. A numerical example and a nonlinear process optimization problem are studied to demonstrate its efficacy. • The distributionally robust chance-constrained programming (DRCCP) optimization based on the kernel ambiguity set is proposed. • The kernel ambiguity set is constructed by utilizing the kernel mean embedding and the maximum mean discrepancy between distributions. • The kernel based DRCCP is formulated as two different models with different approximation and computational complexity. • The kernel based DRCCP is capable of handling general nonlinear optimization problems. [ABSTRACT FROM AUTHOR]

• A data-driven Wasserstein distributionally robust optimization model is proposed. • The day-head scheduling decision of VPP can be solved by off-the-shell solver. • A set of data-driven linearization power flow constraints are constructed. • The model computation efficiency is improved for solving the decisions. Distributed energy resources (DER) can be efficiently aggregated by aggregators to sell excessive electricity to spot market in the form of Virtual Power Plant (VPP). The aggregator schedules DER within VPP to participate in day-ahead market for maximizing its profits while keeping the static operating envelope provided by distribution system operator (DSO) in real-time operation. Aggregator, however, needs to make a decision of its offer for biding under the uncertainties of market price and wind power. This paper proposes a two-stage data-driven scheduling model of VPP in day-ahead (DA) and real time (RT) market. In DA market, in order to determine VPP output for biding, a piece-wise affine formulation of VPP profits combing with CVaR for avoiding market price risk is constructed firstly, and then a data-driven distributionally robust model using a Wasserstein ambiguity set is constructed under uncertainties of market price and wind forecast errors. A set of data-driven linearization power constraints are applied in both DA and RT operation when the parameters of distribution network are unknown or inexact. The model then is reformulated equivalently to a mixed 0–1 convex programming problem. The proposed scheduling model is tested on the IEEE 33-bus distribution network showing that under same 1000-sample dataset in training, proposed DRO model has over 85% of reliability while the stochastic optimization has only 69% under the market risk, which means the proposed model has a better out-of-sample performance for uncertainties. [ABSTRACT FROM AUTHOR]