We develop a novel randomised block coordinate primal-dual algorithm for a class of non-smooth ill-posed convex programs. Lying in the midway between the celebrated Chambolle-Pock primal-dual algorithm and Tseng's accelerated proximal gradient method, we establish global convergence of the last iterate as well optimal $O(1/k)$ and $O(1/k^{2})$ complexity rates in the convex and strongly convex case, respectively, $k$ being the iteration count. Motivated by the increased complexity in the control of distribution level electric power systems, we test the performance of our method on a second-order cone relaxation of an AC-OPF problem. Distributed control is achieved via the distributed locational marginal prices (DLMPs), which are obtained \revise{as} dual variables in our optimisation framework.

翻译：针对一类非光滑不适定凸规划问题，我们提出了一种新颖的随机分块坐标原始-对偶算法。该算法介于著名的Chambolle-Pock原始-对偶算法与Tseng加速近端梯度法之间，我们建立了其最终迭代的全局收敛性，并分别在凸情形和强凸情形下给出了最优的$O(1/k)$和$O(1/k^{2})$复杂度率，其中$k$为迭代次数。受配电级电力系统控制复杂性增加的启发，我们在交流最优潮流问题的二阶锥松弛模型上测试了所提方法的性能。分布式控制通过分布式节点边际电价实现，该电价在我们的优化框架中作为对偶变量获得。