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.

翻译：在线库存优化（OIO）是一种具有物理存储特性的在线凸优化问题：库存结转使得可行动作集依赖于历史决策。在随机库存学习中，以及近期在单一线性容量约束下的OIO问题中，一个自然准则是通过在线学习器维持一个隐目标，并实现其在当前可行补货集上的投影。我们证明，这一简单准则对于任意有界凸容量集上的OIO问题是最优的。当以在线梯度下降为基础学习器时，该方法将一般凸集上OIO问题的最佳已知遗憾界从公共需求概率的逆次改进为逆平方根依赖，并给出了匹配的下界。对于强凸损失函数，该准则首次实现了多对数遗憾保证；对于一般凸容量集上适应欧氏路径变动的动态遗憾，也建立了首个保证。分析引入了一种范数对齐原则：正确的状态变量是隐目标到可行集的距离，且该距离采用与投影相同的范数进行度量。在范数对齐下，该距离沿路径演变为一个标量队列，其中目标移动为到达过程，公共需求为服务过程。这种降维为一维队列控制的方法消除了状态依赖性，并将保证范围扩展至一般凸容量集，突破了以往乘积方法的局限。合成与真实库存数据的实验验证了理论结果。