Optimal Hidden-Target Learning for Online Inventory Optimization on General Convex Sets

Online inventory optimization (OIO) is online convex optimization with physical memory: inventory can be raised immediately, but it can be reduced only by demand. A natural principle, used in stochastic inventory learning and recently in OIO with linear capacity constraints, 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 hidden 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 analysis identifies the right geometric state variable: the Euclidean distance between the hidden target and the implementable set. This distance evolves pathwise as a scalar queue, with target movement as arrivals and common demand as service, reducing the state-dependent feasibility cost to the control of a one-dimensional queue. The same reduction gives the first logarithmic regret guarantee for strongly convex losses and the first dynamic regret guarantee adapting to Euclidean path variation on general convex capacity sets.