This paper considers online convex optimization with long term constraints, where constraints can be violated in intermediate rounds, but need to be satisfied in the long run. The cumulative constraint violation is used as the metric to measure constraint violations, which excludes the situation that strictly feasible constraints can compensate the effects of violated constraints. A novel algorithm is first proposed and it achieves an $\mathcal{O}(T^{\max\{c,1-c\}})$ bound for static regret and an $\mathcal{O}(T^{(1-c)/2})$ bound for cumulative constraint violation, where $c\in(0,1)$ is a user-defined trade-off parameter, and thus has improved performance compared with existing results. Both static regret and cumulative constraint violation bounds are reduced to $\mathcal{O}(\log(T))$ when the loss functions are strongly convex, which also improves existing results. %In order to bound the regret with respect to any comparator sequence, In order to achieve the optimal regret with respect to any comparator sequence, another algorithm is then proposed and it achieves the optimal $\mathcal{O}(\sqrt{T(1+P_T)})$ regret and an $\mathcal{O}(\sqrt{T})$ cumulative constraint violation, where $P_T$ is the path-length of the comparator sequence. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.

翻译：本文以长期限制来考虑在线的 convex优化, 中间回合中可能会违反限制, 但需要长期满足。 累积限制违约被作为衡量限制违约情况的标准, 它排除了严格可行的限制可以补偿被违反限制影响的情况。 首次提出了一个新算法, 它实现了$\ mathcal{O}( T ⁇ max ⁇ c, 1- c ⁇ ) 美元, 用于静态遗憾和$\ mathcal{O} (T ⁇ - P) (1- c)/2} 美元, 用于累计限制违约, 其中$\ in( 0, 1美元) 是用户定义的对交易的参数, 因而比现有结果提高了业绩。 当损失功能强烈的 convex 时, 将静态遗憾和累积限制违约情况都减为$\ mathcal{ (\ log) 美元, 这也会改善现有结果。% 为了约束对任何比较序列的遗憾, 为了实现任何参照序列中的最佳选择, 另一种算法是 $\ true\ cal_ rence_ ral} ral_ cal_ a rock.