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A firm is selling a product to different types (based on the features such as education backgrounds, ages, etc.) of customers over a finite season with non-replenishable initial inventory. The type label of an arriving customer can be observed but the demand function associated with each type is initially unknown. The firm sets personalized prices dynamically for each type and attempts to maximize the revenue over the season. We provide a learning algorithm that is near-optimal when the demand and capacity scale in proportion. The algorithm utilizes the primal-dual formulation of the problem and learns the dual optimal solution explicitly. It allows the algorithm to overcome the curse of dimensionality (the rate of regret is independent of the number of types) and sheds light on novel algorithmic designs for learning problems with resource constraints.

A Primal-dual Learning Algorithm for Personalized Dynamic Pricing with an Inventory Constraint