We propose new algorithms with provable performance for online binary optimization subject to general constraints and in dynamic settings. We consider the subset of problems in which the objective function is submodular. We propose the online submodular greedy algorithm (OSGA) which solves to optimality an approximation of the previous round loss function to avoid the NP-hardness of the original problem. We extend OSGA to a generic approximation function. We show that OSGA has a dynamic regret bound similar to the tightest bounds in online convex optimization with respect to the time horizon and the cumulative round optimum variation. For instances where no approximation exists or a computationally simpler implementation is desired, we design the online submodular projected gradient descent (OSPGD) by leveraging the Lova\'sz extension. We obtain a regret bound that is akin to the conventional online gradient descent (OGD). Finally, we numerically test our algorithms in two power system applications: fast-timescale demand response and real-time distribution network reconfiguration.

翻译：我们提出了具有可证明性能的新算法，用于在动态环境下处理带一般约束的在线二值优化问题。我们考虑目标函数为子模的子问题集。提出了在线子模贪婪算法（OSGA），通过求解前一回合损失函数的最优近似来避免原始问题的NP-hard性。我们将OSGA扩展至通用近似函数，并证明其动态遗憾界在时间跨度和累计回合最优值变化方面与在线凸优化中最紧致的界一致。对于不存在近似函数或需要更简单计算实现的情况，我们利用Lovász扩展设计了在线子模投影梯度下降法（OSPGD），其遗憾界与经典在线梯度下降法（OGD）类似。最后，我们在两个电力系统应用中对算法进行数值测试：快速时间尺度需求响应与实时配电网重构。