Partition-based distributionally robust optimization via optimal transport with order cone constraints

In this paper we wish to tackle stochastic programs affected by ambiguity about the probability law that governs their uncertain parameters. Using optimal transport theory, we construct an ambiguity set that exploits the knowledge about the distribution of the uncertain parameters, which is provided by: (1) sample data and (2) a-priori information on the order among the probabilities that the true data-generating distribution assigns to some regions of its support set. This type of order is enforced by means of order cone constraints and can encode a wide range of information on the shape of the probability distribution of the uncertain parameters such as information related to monotonicity or multi-modality. We seek decisions that are distributionally robust. In a number of practical cases, the resulting distributionally robust optimization (DRO) problem can be reformulated as a finite convex problem where the a-priori information translates into linear constraints. In addition, our method inherits the finite-sample performance guarantees of the Wasserstein-metric-based DRO approach proposed by Mohajerin Esfahani and Kuhn (Math Program 171(1–2):115–166. https://doi.org/10.1007/s10107-017-1172-1, 2018), while generalizing this and other popular DRO approaches. Finally, we have designed numerical experiments to analyze the performance of our approach with the newsvendor problem and the problem of a strategic firm competing à la Cournot in a market.This research has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 755705). This work was also supported in part by the Spanish Ministry of Economy, Industry and Competitiveness and the European Regional Development Fund (ERDF) through Project ENE2017-83775-P

Conic Reformulations for Kullback-Leibler Divergence Constrained Distributionally Robust Optimization and Applications

In this paper, we consider a distributionally robust optimization (DRO) model in which the ambiguity set is defined as the set of distributions whose Kullback-Leibler (KL) divergence to an empirical distribution is bounded. Utilizing the fact that KL divergence is an exponential cone representable function, we obtain the robust counterpart of the KL divergence constrained DRO problem as a dual exponential cone constrained program under mild assumptions on the underlying optimization problem. The resulting conic reformulation of the original optimization problem can be directly solved by a commercial conic programming solver. We specialize our generic formulation to two classical optimization problems, namely, the Newsvendor Problem and the Uncapacitated Facility Location Problem. Our computational study in an out-of-sample analysis shows that the solutions obtained via the DRO approach yield significantly better performance in terms of the dispersion of the cost realizations while the central tendency deteriorates only slightly compared to the solutions obtained by stochastic programming

Distribution-free Inventory Risk Pooling in a Multi-location Newsvendor

With rapidly increasing e-commerce sales, firms are leveraging the virtual pooling of online demands across customer locations in deciding the amount of inventory to be placed in each node in a fulfillment network. Such solutions require knowledge of the joint distribution of demands; however, in reality, only partial information about the joint distribution may be reliably estimated. We propose a distributionally robust multi-location newsvendor model for network inventory optimization where the worst-case expected cost is minimized over the set of demand distributions satisfying the known mean and covariance information. For the special case of two homogeneous customer locations with correlated demands, we show that a six-point distribution achieves the worst-case expected cost, and derive a closed-form expression for the optimal inventory decision. The general multi-location problem can be shown to be NP-hard. We develop a computationally tractable upper bound through the solution of a semidefinite program (SDP), which also yields heuristic inventory levels, for a special class of fulfillment cost structures, namely nested fulfillment structures. We also develop an algorithm to convert any general distance-based fulfillment cost structure into a nested fulfillment structure which tightly approximates the expected total fulfillment cost.https://deepblue.lib.umich.edu/bitstream/2027.42/146785/1/1389_Govindarajan.pd

Data-driven stochastic optimization for distributional ambiguity with integrated confidence region

We discuss stochastic optimization problems under distributional ambiguity. The distributional uncertainty is captured by considering an entire family of distributions. Because we assume the existence of data, we can consider confidence regions for the different estimators of the parameters of the distributions. Based on the definition of an appropriate estimator in the interior of the resulting confidence region, we propose a new data-driven stochastic optimization problem. This new approach applies the idea of a-posteriori Bayesian methods to the confidence region. We are able to prove that the expected value, over all observations and all possible distributions, of the optimal objective function of the proposed stochastic optimization problem is bounded by a constant. This constant is small for a sufficiently large i.i.d. sample size and depends on the chosen confidence level and the size of the confidence region. We demonstrate the utility of the new optimization approach on a Newsvendor and a reliability problem

Distributionally Robust Optimization: A Review

The concepts of risk-aversion, chance-constrained optimization, and robust optimization have developed significantly over the last decade. Statistical learning community has also witnessed a rapid theoretical and applied growth by relying on these concepts. A modeling framework, called distributionally robust optimization (DRO), has recently received significant attention in both the operations research and statistical learning communities. This paper surveys main concepts and contributions to DRO, and its relationships with robust optimization, risk-aversion, chance-constrained optimization, and function regularization