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This paper deals with the problem of quantifying the impact of model misspecification when computing general expected values of interest. The methodology that we propose is applicable in great generality; in particular, we provide examples involving path-dependent expectations of stochastic processes. Our approach consists of computing bounds for the expectation of interest regardless of the probability measure used, as long as the measure lies within a prescribed tolerance measured in terms of a flexible class of distances from a suitable baseline model. These distances, based on optimal transportation between probability measures, include Wasserstein’s distances as particular cases. The proposed methodology is well suited for risk analysis and distributionally robust optimization, as we demonstrate with applications. We also discuss how to estimate the tolerance region nonparametrically using Skorokhod-type embeddings in some of these applications.

We present a new distributionally robust optimization model called robust stochastic optimization (RSO), which unifies both scenario-tree-based stochastic linear optimization and distributionally robust optimization in a practicable framework that can be solved using the state-of-the-art commercial optimization solvers. We also develop a new algebraic modeling package, Robust Stochastic Optimization Made Easy (RSOME), to facilitate the implementation of RSO models. The model of uncertainty incorporates both discrete and continuous random variables, typically assumed in scenario-tree-based stochastic linear optimization and distributionally robust optimization, respectively. To address the nonanticipativity of recourse decisions, we introduce the event-wise recourse adaptations, which integrate the scenario-tree adaptation originating from stochastic linear optimization and the affine adaptation popularized in distributionally robust optimization. Our proposed event-wise ambiguity set is rich enough to capture traditional statistic-based ambiguity sets with convex generalized moments, mixture distribution, φ-divergence, Wasserstein (Kantorovich-Rubinstein) metric, and also inspire machine-learning-based ones using techniques such as K-means clustering and classification and regression trees. Several interesting RSO models, including optimizing over the Hurwicz criterion and two-stage problems over Wasserstein ambiguity sets, are provided. This paper was accepted by David Simchi-Levi, optimization.

Adaptive robust optimization problems are usually solved approximately by restricting the adaptive decisions to simple parametric decision rules. However, the corresponding approximation error can be substantial. In this paper we show that two-stage robust and distributionally robust linear programs can often be reformulated exactly as conic programs that scale polynomially with the problem dimensions. Specifically, when the ambiguity set constitutes a 2-Wasserstein ball centered at a discrete distribution, the distributionally robust linear program is equivalent to a copositive program (if the problem has complete recourse) or can be approximated arbitrarily closely by a sequence of copositive programs (if the problem has sufficiently expensive recourse). These results directly extend to the classical robust setting and motivate strong tractable approximations of two-stage problems based on semidefinite approximations of the copositive cone. We also demonstrate that the two-stage distributionally robust optimization problem is equivalent to a tractable linear program when the ambiguity set constitutes a 1-Wasserstein ball centered at a discrete distribution and there are no support constraints. The online appendix is available at https://doi.org/10.1287/opre.2017.1698 .

We show that several machine learning estimators, including square-root least absolute shrinkage and selection and regularized logistic regression, can be represented as solutions to distributionally robust optimization problems. The associated uncertainty regions are based on suitably defined Wasserstein distances. Hence, our representations allow us to view regularization as a result of introducing an artificial adversary that perturbs the empirical distribution to account for out-of-sample effects in loss estimation. In addition, we introduce RWPI (robust Wasserstein profile inference), a novel inference methodology which extends the use of methods inspired by empirical likelihood to the setting of optimal transport costs (of which Wasserstein distances are a particular case). We use RWPI to show how to optimally select the size of uncertainty regions, and as a consequence we are able to choose regularization parameters for these machine learning estimators without the use of cross validation. Numerical experiments are also given to validate our theoretical findings.

Sample average approximation (SAA) is a widely popular approach to data-driven decision-making under uncertainty. Under mild assumptions, SAA is both tractable and enjoys strong asymptotic performance guarantees. Similar guarantees, however, do not typically hold in finite samples. In this paper, we propose a modification of SAA, which we term Robust SAA, which retains SAA’s tractability and asymptotic properties and, additionally, enjoys strong finite-sample performance guarantees. The key to our method is linking SAA, distributionally robust optimization, and hypothesis testing of goodness-of-fit. Beyond Robust SAA, this connection provides a unified perspective enabling us to characterize the finite sample and asymptotic guarantees of various other data-driven procedures that are based upon distributionally robust optimization. This analysis provides insight into the practical performance of these various methods in real applications. We present examples from inventory management and portfolio allocation, and demonstrate numerically that our approach outperforms other data-driven approaches in these applications.

Distributionally robust optimization (DRO), a recent methodology to handle stochastic optimization problems in the presence of data, is based on robustifications of stochastic constraints that are enforced to hold over suitably constructed sets of underlying probability distributions. Although DRO enjoys valid feasibility guarantees, it often leads to over-conservative solutions. The paper “Recovering best statistical guarantees via the empirical divergence-based distributionally robust optimization” by Lam studies a calibration method for distributional sets to combat conservativeness via a new interpretation of DRO through the statistical angle of empirical likelihood and empirical processes. The proposed method targets achieving precise confidence level guarantees that lead to superior performances over previous approaches. 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 because, 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 χ 2 -process excursion. We conduct numerical experiments to support our theoretical findings.

A coordinated scheduling model based on two-stage distributionally robust optimization (TSDRO) is proposed for integrated energy systems (IESs) with electricity-hydrogen hybrid energy storage. The scheduling problem of the IES is divided into two stages in the TSDRO-based coordinated scheduling model. The first stage addresses the day-ahead optimal scheduling problem of the IES under deterministic forecasting information, while the second stage uses a distributionally robust optimization method to determine the intraday rescheduling problem under high-order uncertainties, building upon the results of the first stage. The scheduling model also considers collaboration among the electricity, thermal, and gas networks, focusing on economic operation and carbon emissions. The flexibility of these networks and the energy gradient utilization of hydrogen units during operation are also incorporated into the model. To improve computational efficiency, the nonlinear formulations in the TSDRO-based coordinated scheduling model are properly linearized to obtain a Mixed-Integer Linear Programming model. The Column-Constraint Generation (C&CG) algorithm is then employed to decompose the scheduling model into a master problem and subproblems. Through the iterative solution of the master problem and subproblems, an efficient analysis of the coordinated scheduling model is achieved. Finally, the effectiveness of the proposed TSDRO-based coordinated scheduling model is verified through case studies. The simulation results demonstrate that the proposed TSDRO-based coordinated scheduling model can effectively accomplish the optimal scheduling task while considering the uncertainty and flexibility of the system. Compared with traditional methods, the proposed TSDRO-based coordinated scheduling model can better balance conservativeness and robustness.

This paper studies how to schedule medical appointments with time-dependent patient no-show behavior and random service times. The problem is motivated by our studies of independent datasets from countries in two continents that unanimously identify a significant time-of-day effect on patient show-up probabilities. We deploy a distributionally robust model, which minimizes the worst-case total expected costs of patient waiting and service provider’s idling and overtime, by optimizing the scheduled arrival times of patients. This model is challenging because evaluating the total cost for a given schedule involves a linear program with uncertainties present in both the objective function and the right-hand side of the constraints. In addition, the ambiguity set considered contains discrete uncertainties and complementary functional relationships among these uncertainties (namely, patient no-shows and service durations). We show that when patient no-shows are exogenous (i.e., time-independent), the problem can be reformulated as a copositive program and then be approximated by semidefinite programs. When patient no-shows are endogenous on time (and hence on the schedule), the problem becomes a bilinear copositive program. We construct a set of dual prices to guide the search for a good schedule and use the technique iteratively to obtain a near-optimal solution. Our computational studies reveal a significant reduction in total expected cost by taking into account the time-of-day variation in patient show-up probabilities as opposed to ignoring it. This paper was accepted by David Simchi-Levi, optimization.

We study joint chance constraints where the distribution of the uncertain parameters is only known to belong to an ambiguity set characterized by the mean and support of the uncertainties and by an upper bound on their dispersion. This setting gives rise to pessimistic (optimistic) ambiguous chance constraints, which require the corresponding classical chance constraints to be satisfied for every (for at least one) distribution in the ambiguity set. We demonstrate that the pessimistic joint chance constraints are conic representable if (i) the constraint coefficients of the decisions are deterministic, (ii) the support set of the uncertain parameters is a cone, and (iii) the dispersion function is of first order, that is, it is positively homogeneous. We also show that pessimistic joint chance constrained programs become intractable as soon as any of the conditions (i), (ii) or (iii) is relaxed in the mildest possible way. We further prove that the optimistic joint chance constraints are conic representable if (i) holds, and that they become intractable if (i) is violated. We show in numerical experiments that our results allow us to solve large-scale project management and image reconstruction models to global optimality. The online appendix is available at https://doi.org/10.1287/opre.2016.1583 .

We study distributionally robust chance-constrained programming (DRCCP) optimization problems with data-driven Wasserstein ambiguity sets. The proposed algorithmic and reformulation framework applies to all types of distributionally robust chance-constrained optimization problems subjected to individual as well as joint chance constraints, with random right-hand side and technology vector, and under two types of uncertainties, called uncertain probabilities and continuum of realizations. For the uncertain probabilities (UP) case, we provide new mixed-integer linear programming reformulations for DRCCP problems. For the continuum of realizations case with random right-hand side, we propose an exact mixed-integer second-order cone programming (MISOCP) reformulation and a linear programming (LP) outer approximation. For the continuum of realizations (CR) case with random technology vector, we propose two MISOCP and LP outer approximations. We show that all proposed relaxations become exact reformulations when the decision variables are binary or bounded general integers. For DRCCP with individual chance constraint and random right-hand side under both the UP and CR cases, we also propose linear programming reformulations which need the ex-ante derivation of the worst-case value-at-risk via the solution of a finite series of linear programs determined via a bisection-type procedure. We evaluate the scalability and tightness of the proposed MISOCP and (MI)LP formulations on a distributionally robust chance-constrained knapsack problem.