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Chasing Convex Functions with Long-term Constraints

Abstract

We introduce and study a family of online metric problems with long-term constraints. In these problems, an online player makes decisions xt\mathbf{x}_t in a metric space (X,d)(X,d) to simultaneously minimize their hitting cost ft(xt)f_t(\mathbf{x}_t) and switching cost as determined by the metric. Over the time horizon TT, the player must satisfy a long-term demand constraint tc(xt)1\sum_{t} c(\mathbf{x}_t) \geq 1, where c(xt)c(\mathbf{x}_t) denotes the fraction of demand satisfied at time tt. Such problems can find a wide array of applications to online resource allocation in sustainable energy/computing systems. We devise optimal competitive and learning-augmented algorithms for the case of bounded hitting cost gradients and weighted 1\ell_1 metrics, and further show that our proposed algorithms perform well in numerical experiments.

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