Online Linear Programming with Replenishment

We study an online linear programming (OLP) model in which inventory is not provided upfront but instead arrives gradually through an exogenous stochastic replenishment process. This replenishment-based formulation captures operational settings, such as e-commerce fulfillment, perishable supply chains, and renewable-powered systems, where resources are accumulated gradually and initial inventories are small or zero. The introduction of dispersed, uncertain replenishment fundamentally alters the structure of classical OLPs, creating persistent stockout risk and eliminating advance knowledge of the total budget. We develop new algorithms and regret analyses for three major distributional regimes studied in the OLP literature: bounded distributions, finite-support distributions, and continuous-support distributions with a non-degeneracy condition. For bounded distributions, we design an algorithm that achieves $\widetilde{\mathcal{O}}(\sqrt{T})$ regret. For finite-support distributions with a non-degenerate induced LP, we obtain $\mathcal{O}(\log T)$ regret, and we establish an $Ω(\sqrt{T})$ lower bound for degenerate instances, demonstrating a sharp separation from the classical setting where $\mathcal{O}(1)$ regret is achievable. For continuous-support, non-degenerate distributions, we develop a two-stage accumulate-then-convert algorithm that achieves $\mathcal{O}(\log^2 T)$ regret, comparable to the $\mathcal{O}(\log T)$ regret in classical OLPs. Together, these results provide a near-complete characterization of the optimal regret achievable in OLP with replenishment. Finally, we empirically evaluate our algorithms and demonstrate their advantages over natural adaptations of classical OLP methods in the replenishment setting.

A Minimax-MDP Framework with Future-imposed Conditions for Learning-augmented Problems

We study a class of sequential decision-making problems with augmented predictions, potentially provided by a machine learning algorithm. In this setting, the decision-maker receives prediction intervals for unknown parameters that become progressively refined over time, and seeks decisions that are competitive with the hindsight optimal under all possible realizations of both parameters and predictions. We propose a minimax Markov Decision Process (minimax-MDP) framework, where the system state consists of an adversarially evolving environment state and an internal state controlled by the decision-maker. We introduce a set of future-imposed conditions that characterize the feasibility of minimax-MDPs and enable the design of efficient, often closed-form, robustly competitive policies. We illustrate the framework through three applications: multi-period inventory ordering with refining demand predictions, resource allocation with uncertain utility functions, and a multi-phase extension of the minimax-MDP applied to the inventory problem with time-varying ordering costs. Our results provide a tractable and versatile approach to robust online decision-making under predictive uncertainty.