We study multi-period risk-averse inventory control in a data-driven setting. In this problem, a risk-averse retailer makes periodic decisions on inventory levels based only on historical demand observations without full knowledge of the demand distribution. We adopt the popular nested formulation for risk-averse programs to formulate this multi-period problem and its data-driven counterpart under a coherent risk measure. Our objective is to study the sample complexity bound such that with high probability, the data-driven policy is near-optimal, that is, the relative error of risk under the data-driven policy compared with the optimal risk is arbitrarily small. Analysis of this problem is inherently challenging, because the multi-period nature requires solving the risk-averse program and its data-driven version recursively backward in time, while the (empirical) risk-to-go functions in this process do not have closed-form derivatives for most risk measures, which renders existing first-order methods for the risk-neutral newsvendor model invalid. In this study, we develop a zeroth-order framework to establish the complexity bound on sample sizes to guarantee near-optimality of the data-driven policy with given accuracy levels. Instead of using first-order derivative information on the risk-to-go function, our analysis directly examines the class of functions that underpins each cumulative risk function and derives maximum inequalities for this functional class by computing the covering numbers. Finite-sample complexity bounds are then used to establish asymptotic properties of the estimated risk, including consistency and convergence rate. Computationally, the time complexity for solving the data-driven policy, which is essentially an empirical dynamic programming (EDP) estimator of the optimal policy, increases exponentially in the length of the planning horizon. To speed up computation, we propose an approximation scheme that recursively approximates the empirical cumulative risk function with a convex piecewise linear function and then minimize it to obtain a modified data-driven inventory policy. We show that with proper control for approximation error, the modified data-driven policy is also near-optimal, and it has the same order of sample complexity bound as that for the original EDP policy.

Problem definition: Normal operations of a power system require that alternating current frequency be maintained at a nominal value, for example, 50 Hz, whereas severe deviation from this value due to power deficiencies can cause cascading generator trips. Maintaining the frequency requires adequate inertia and frequency regulation reserve, which are primarily provided by online generators. In daily operations, generators due for preventive maintenance must be taken offline, and thus an improper maintenance schedule could jeopardize frequency security, as exemplified by the recent Texas power blackout. However, this natural nexus between frequency security and maintenance has been overlooked largely in the literature. Methodology/results: We fill the gap by developing a long-term generator maintenance scheduling model that incorporates frequency security constraints with hourly fidelity to meet industrial standards. These constraints amount to scheduling adequate inertia and frequency regulation reserve by considering uncertain power deficiency and inertia from intermittent renewable energy. We hedge the uncertainties by employing a robust optimization approach in which historical data are used to construct ambiguity sets. This inevitably results in an ultra-large-scale robust model because of the hourly fidelity. We reformulate it as a large-scale, mixed-integer linear program. An algorithm based on the progressive hedging idea is proposed to decompose the model into subprograms that can be solved in parallel. An explicit-dual cutting-plane method for the subprograms and a novel lower bound for the model are developed to accelerate computation in each iteration. Compared with the standard progressive hedging algorithm and an L-shaped algorithm with strengthened Benders cuts, our algorithm is approximately 10 times faster and avoids the out-of-memory issues encountered by these benchmarks. Managerial implications: Integrating frequency security enforces generator maintenance to distribute more evenly across the planning horizon. This leads to a more stable maintenance crew size and a significant reduction in out-of-sample costs in our simulation using real data. Additionally, our study reveals that inertia is crucial for frequency security and that low-cost inertia resources like synchronous condensers can enhance frequency security. Funding: The research was conducted at the University of Macau, supported by the UM Grant SRG2025-00044-IOTSC and by FDCT support 001/2024/SKL (Y. Yang). This research was supported by the National Science Foundation of China 72471144 (Q. Sun). This research was supported by Singapore MOE AcRF Tier 2 Grant [A-8001052-00-00, A-8002472-00-00] (Z. Ye). The research was conducted at the Future Resilient Systems at the Singapore-ETH Centre, which was established collaboratively between ETH Zurich and the National Research Foundation Singapore. This research is supported by the National Research Foundation Singapore (NRF) under its Campus for Research Excellence and Technological Enterprise (CREATE) programme (J.C.-H. Peng, L.C. Tang, Z. Ye). Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2023.0664 .