Chasing Convex Functions with Long-term Constraints
Adam Lechowicz
Nicolas H. Christianson
Bo Sun
Noman Bashir
Mohammad Hajiesmaili
Adam Wierman
Prashant J. Shenoy

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