Competitive Algorithms for Online Budget-Constrained Continuous DR-Submodular Problems

In this paper, we study a certain class of online optimization problems, where the goal is to maximize a function that is not necessarily concave and satisfies the Diminishing Returns (DR) property under budget constraints. We analyze a primal-dual algorithm, called the Generalized Sequential algorithm, and we obtain the first bound on the competitive ratio of online monotone DR-submodular function maximization subject to linear packing constraints which matches the known tight bound in the special case of linear objective function.

Exponentiated Subgradient Algorithm for Online Optimization under the Random Permutation Model

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 $1-O(\epsilon)$ under a near-optimal assumption that the bid to budget ratio is $O (\frac{\epsilon^2}{\log({m}/{\epsilon})})$, 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.