Simultaneously Achieving Sublinear Regret and Constraint Violations for Online Convex Optimization with Time-varying Constraints

In this paper, we develop a novel virtual-queue-based online algorithm for online convex optimization (OCO) problems with long-term and time-varying constraints and conduct a performance analysis with respect to the dynamic regret and constraint violations. We design a new update rule of dual variables and a new way of incorporating time-varying constraint functions into the dual variables. To the best of our knowledge, our algorithm is the first parameter-free algorithm to simultaneously achieve sublinear dynamic regret and constraint violations. Our proposed algorithm also outperforms the state-of-the-art results in many aspects, e.g., our algorithm does not require the Slater condition. Meanwhile, for a group of practical and widely-studied constrained OCO problems in which the variation of consecutive constraints is smooth enough across time, our algorithm achieves O(1) constraint violations. Furthermore, we extend our algorithm and analysis to the case when the time horizon T is unknown. Finally, numerical experiments are conducted to validate the theoretical guarantees of our algorithm, and some applications of our proposed framework will be outlined.