A novel and fully distributed optimization method is proposed for the distributed robust convex program (DRCP) over a time-varying unbalanced directed network without imposing any differentiability assumptions. Firstly, a tractable approximated DRCP (ADRCP) is introduced by discretizing the semi-infinite constraints into a finite number of inequality constraints and restricting the right-hand side of the constraints with a proper positive parameter, which will be iteratively solved by a random-fixed projection algorithm. Secondly, a cutting-surface consensus approach is proposed for locating an approximately optimal consensus solution of the DRCP with guaranteed feasibility. This approach is based on iteratively approximating the DRCP by successively reducing the restriction parameter of the right-hand constraints and populating the cutting-surfaces into the existing finite set of constraints. Thirdly, to ensure finite-time convergence of the distributed optimization, a distributed termination algorithm is developed based on uniformly local consensus and zeroth-order optimality under uniformly strongly connected graphs. Fourthly, it is proved that the cutting-surface consensus approach converges within a finite number of iterations. Finally, the effectiveness of the approach is illustrated through a numerical example.

翻译：针对时变不平衡有向网络上的分布式鲁棒凸规划（DRCP），本文提出了一种新颖且完全分布式的优化方法，该方法无需任何可微性假设。首先，通过将半无限约束离散化为有限数量的不等式约束，并使用适当的正参数限制约束的右侧，引入了一种可处理的近似DRCP（ADRCP），该问题将通过随机固定投影算法迭代求解。其次，提出了一种切割面共识方法，用于寻找具有可行性保证的DRCP近似最优共识解。该方法通过逐步减小右侧约束的限制参数，并将切割面填充到现有有限约束集中，来迭代逼近DRCP。第三，为确保分布式优化的有限时间收敛，基于一致局部共识和零阶最优性，在一致强连通图下开发了一种分布式终止算法。第四，证明了切割面共识方法在有限迭代次数内收敛。最后，通过数值示例验证了该方法的有效性。