We introduce and study a family of online metric problems with long-term
constraints. In these problems, an online player makes decisions $\mathbf{x}_t$
in a metric space $(X,d)$ to simultaneously minimize their hitting cost
$f_t(\mathbf{x}_t)$ and switching cost as determined by the metric. Over the
time horizon $T$, the player must satisfy a long-term demand constraint
$\sum_{t} c(\mathbf{x}_t) \geq 1$, where $c(\mathbf{x}_t)$ 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 and computing
systems. We devise optimal competitive and learning-augmented algorithms for
specific instantiations of these problems, and further show that our proposed
algorithms perform well in numerical experiments.

我们研究了一类带有长期约束的在线度量问题，该问题涉及在线玩家在度量空间中进行决策以同时最小化击中成本和度量确定的切换成本。我们设计了特定实例的最优竞争性和学习增强算法，并进一步在数值实验中证明了我们提出的算法的良好性能。