Resource-Adaptive Primal-Dual Learning for One-Warehouse Multi-Store Systems with Censored Demand

Many real-world resource allocation systems, such as humanitarian logistics and vaccine distribution, must preposition limited supply across multiple locations before demand is realized while stockouts incur irreversible service losses. To study this, we introduce the Online Shared Supply Allocation (OSSA) problem, a stateful online model in which a central hub allocates a finite, unknown supply to multiple sites facing sequential demand under fixed-charge transportation costs and lost-sales penalties. Unlike classical make-to-stock or make-to-order inventory models, OSSA precludes backlogging and replenishment only hedges against future demand. To tackle OSSA, we propose a deterministic threshold-proportional policy GPA and prove that it achieves a $4/3$-approximation to the offline optimum up to an additive term independent of the total supply. We complement this with matching lower bounds showing that the $4/3$ ratio is tight and that the additive-error dependence is unavoidable, even for randomized algorithms that know the total supply upfront. Finally, we develop a learning-augmented extension to GPA that principally incorporates imperfect forecasts (e.g., from human experts or ML models) commonly available in practice, enabling us to exploit high-quality advice while being robust against arbitrary bad ones. Synthetic and real-world experiments show that GPA outperforms natural baselines with global supply is scarce.

We study a finite-horizon online resource allocation problem with initial resource capacities proportional to the horizon. In each period, a request type is observed and one action is chosen from a finite menu. Each action earns a reward and consumes a vector of resources. We present a primal first-order learning policy that, in each period, performs one gradient ascent update of the action coordinates associated with one request type, for a fluid relaxation in which resource overuse incurs a quadratic penalty. The policy achieves $O(1)$ expected additive regret relative to the hindsight optimum, with a bound independent of the horizon $T$, under a general arrival model. This model includes independent and identically distributed arrivals with unknown probabilities, independent nonstationary arrivals with known probabilities, independent nonstationary arrivals with unknown probabilities and one historical sample path, and Markov-modulated arrivals with an unknown Markov chain. The policy does not solve any linear program, and the regret bound does not require a nondegeneracy assumption on the fluid linear program.

Online inventory optimization (OIO) is online convex optimization with physical memory: inventory carryover makes the feasible action set depend on the past. A natural principle, used in stochastic inventory learning and recently in OIO under a single linear capacity constraint, is to maintain a hidden target chosen by an online learner and implement its projection onto the currently feasible order-up-to set. We prove that this simple principle is optimal for OIO on arbitrary bounded convex capacity sets. With online gradient descent as the base learner, the method improves the best known regret guarantee for OIO on general convex sets from inverse to inverse-square-root dependence on the common-demand probability, and we prove a matching lower bound. The same principle gives the first polylogarithmic regret guarantee for strongly convex losses and the first dynamic regret guarantee adapting to Euclidean path variation on general convex capacity sets. The analysis introduces a norm alignment principle: the right state variable is the distance from the hidden target to the feasible set, measured in the same norm as the projection. Under norm alignment, this distance evolves pathwise as a scalar queue, with target movement as arrival and common demand as service. This reduction to one-dimensional queue control resolves the state dependence and extends the guarantees to general convex capacity sets, beyond the reach of prior productwise approaches. Experiments on synthetic and real-world inventory data corroborate the theory.