  Online allocation problems with resource constraints have a rich history in
operations research. In this paper, we introduce the \emph{regularized online
allocation problem}, a variant that includes a non-linear regularizer acting on
the total resource consumption. In this problem, requests repeatedly arrive
over time and, for each request, a decision maker needs to take an action that
generates a reward and consumes resources. The objective is to simultaneously
maximize additively separable rewards and the value of a non-separable
regularizer subject to the resource constraints. Our primary motivation is
allowing decision makers to trade off separable objectives such as the economic
efficiency of an allocation with ancillary, non-separable objectives such as
the fairness or equity of an allocation. We design an algorithm that is simple,
fast, and attains good performance with both stochastic i.i.d.~and adversarial
inputs. In particular, our algorithm is asymptotically optimal under stochastic
i.i.d. input models and attains a fixed competitive ratio that depends on the
regularizer when the input is adversarial. Furthermore, the algorithm and
analysis do not require convexity or concavity of the reward function and the
consumption function, which allows more model flexibility. Numerical
experiments confirm the effectiveness of the proposed algorithm and of
regularization in an internet advertising application.
