Technical Note—Two-Stage Sample Robust Optimization

In “Two-Stage Sample Robust Optimization,” Bertsimas, Shtern, and Sturt investigate a simple approximation scheme, based on overlapping linear decision rules, for solving data-driven two-stage distributionally robust optimization problems with the type-infinity Wasserstein ambiguity set. Their main result establishes that this approximation scheme is asymptotically optimal for two-stage stochastic linear optimization problems; that is, under mild assumptions, the optimal cost and optimal first-stage decisions obtained by approximating the robust optimization problem converge to those of the underlying stochastic problem as the number of data points grows to infinity. These guarantees notably apply to two-stage stochastic problems that do not have relatively complete recourse, which arise frequently in applications. In this context, the authors show through numerical experiments that the approximation scheme is practically tractable and produces decisions that significantly outperform those obtained from state-of-the-art data-driven alternatives.

Robust Stochastic Optimization Made Easy with RSOME

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.

Demand Response of an Industrial Buyer considering Congestion and LMP in Day-Ahead Electricity Market

This paper proposes a methodology for developing Price Responsive Demand Shifting (PRDS) based bidding strategy of an industrial buyer, who can reschedule its production plan, considering power system network constraints. Locational Marginal Price (LMP) methodology, which is being used in PJM, California, New York, and New England electricity markets, has been utilized to manage the congestion. In this work, a stochastic linear optimization formulation comprising of two sub-problems has been proposed to obtain the optimal bidding strategy of an industrial buyer considering PRDS bidding. The first sub-problem is formulated as to maximize the social welfare of market participants subject to operational constraints and security constraints to facilitate market clearing process, while the second sub-problem represents the industrial buyer’s purchase cost saving maximization. The PRDS based bidding strategy, which is able to shift the demand, from high price periods to low price periods, has been obtained by solving two subproblems. The effectiveness of the proposed method has been tested on a 5-bus system and modified IEEE 30-bus system considering the hourly day-ahead market. Results obtained with the PRDS based bidding strategy have been compared with those obtained with a Conventional Price-Quantity (CPQ) bid. In simulation studies, it is observed that the PRDS approach can control the LMPs and congestion at the system buses. It is also found that PRDS can mitigate the market power by flattening the demand, which led to more saving and satisfying demand.