  Online allocation problems with resource constraints are central problems in
revenue management and online advertising. In these problems, requests arrive
sequentially during a finite horizon and, for each request, a decision maker
needs to choose an action that consumes a certain amount of resources and
generates reward. The objective is to maximize cumulative rewards subject to a
constraint on the total consumption of resources. In this paper, we consider a
data-driven setting in which the reward and resource consumption of each
request are generated using an input model that is unknown to the decision
maker. We design a general class of algorithms that attain good performance in
various input models without knowing which type of input they are facing. In
particular, our algorithms are asymptotically optimal under independent and
identically distributed inputs as well as various non-stationary stochastic
input models, and they attain an asymptotically optimal fixed competitive ratio
when the input is adversarial. Our algorithms operate in the Lagrangian dual
space: they maintain a dual multiplier for each resource that is updated using
online mirror descent. By choosing the reference function accordingly, we
recover the dual sub-gradient descent and dual multiplicative weights update
algorithm. The resulting algorithms are simple, fast, and do not require
convexity in the revenue function, consumption function and action space, in
contrast to existing methods for online allocation problems. We discuss
applications to network revenue management, online bidding in repeated auctions
with budget constraints, online proportional matching with high entropy, and
personalized assortment optimization with limited inventory.
