Online optimization problems arise in many resource allocation tasks, where
the future demands for each resource and the associated utility functions
change over time and are not known apriori, yet resources need to be allocated
at every point in time despite the future uncertainty. In this paper, we
consider online optimization problems with general concave utilities. We modify
and extend an online optimization algorithm proposed by Devanur et al. for
linear programming to this general setting. The model we use for the arrival of
the utilities and demands is known as the random permutation model, where a
fixed collection of utilities and demands are presented to the algorithm in
random order. We prove that under this model the algorithm achieves a
competitive ratio of 1−O(ϵ) under a near-optimal assumption that the
bid to budget ratio is O(log(m/ϵ)ϵ2), where m
is the number of resources, while enjoying a significantly lower computational
cost than the optimal algorithm proposed by Kesselheim et al. We draw a
connection between the proposed algorithm and subgradient methods used in
convex optimization. In addition, we present numerical experiments that
demonstrate the performance and speed of this algorithm in comparison to
existing algorithms