Worst case competitive analysis for online conic optimization

We provide a worst-case competitive analysis for two types of primal-dual greedy algorithms for a broad class of online conic optimization problems. This class contains problems such as online resource allocation, online routing in networks, online bipartite matching, Adwords, and Adwords with concave returns. One algorithm updates the primal and dual variables sequentially at each step, while the other algorithm updates them simultaneously. We derive a sufficient condition on the objective function that leads to a bound on the competitive ratio (using the ratio of the objective function to its Fenchel conjugate, which can be seen as a measure of “curvature”). We show how Nesterov's smoothing technique can be utilized to improve the competitive ratio, and in case of online Linear Programs, provide the optimal competitive ratio. We apply our results to a graph formation problem as a new example on the positive semidefinite cone that satisfies our sufficient condition.