Chasing Convex Functions with Long-term Constraints

We introduce and study a family of online metric problems with long-term constraints. In these problems, an online player makes decisions xt in a metric space (X,d) to simultaneously minimize their hitting cost ft(xt) and switching cost as determined by the metric. Over the time horizon T, the player must satisfy a long-term demand constraint sumt c(xt) ≥ 1, where c(xt) denotes the fraction of demand satisfied at time t. 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 ℓ1 metrics, and further show that our proposed algorithms perform well in numerical experiments.