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Online optimization problems arise in many resource allocation tasks, wherethe future demands for each resource and the associated utility functionschange over time and are not known apriori, yet resources need to be allocatedat every point in time despite the future uncertainty. In this paper, weconsider online optimization problems with general concave utilities. We modifyand extend an online optimization algorithm proposed by Devanur et al. forlinear programming to this general setting. The model we use for the arrival ofthe utilities and demands is known as the random permutation model, where afixed collection of utilities and demands are presented to the algorithm inrandom order. We prove that under this model the algorithm achieves acompetitive ratio of $1-O(\epsilon)$ under a near-optimal assumption that thebid to budget ratio is $O (\frac{\epsilon^2}{\log({m}/{\epsilon})})$, where $m$is the number of resources, while enjoying a significantly lower computationalcost than the optimal algorithm proposed by Kesselheim et al. We draw aconnection between the proposed algorithm and subgradient methods used inconvex optimization. In addition, we present numerical experiments thatdemonstrate the performance and speed of this algorithm in comparison toexisting algorithms.

Exponentiated Subgradient Algorithm for Online Optimization under the Random Permutation Model