A novel and fully distributed optimization method is proposed for the distributed robust convex program (DRCP) over a time-varying unbalanced directed network under the uniformly jointly strongly connected (UJSC) assumption. 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 positive parameter. This problem is iteratively solved by a distributed projected gradient algorithm proposed in this paper, which is based on epigraphic reformulation and subgradient projected algorithms. 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 termination of the distributed optimization, a distributed termination algorithm is developed based on consensus and zeroth-order stopping conditions under UJSC graphs. Fourthly, it is proved that the cutting-surface consensus approach terminates finitely and yields a feasible and approximate optimal solution for each agent. Finally, the effectiveness of the approach is illustrated through a numerical example.

翻译：本文针对一致联合强连通（UJSC）假设下时变非平衡有向网络上的分布式鲁棒凸规划（DRCP）问题，提出了一种新颖且完全分布式的优化方法。首先，通过将半无限约束离散化为有限个不等式约束，并用正参数限制约束右侧，引入了一个可处理的近似DRCP（ADRCP）。该问题通过本文提出的分布式投影梯度算法迭代求解，该算法基于epigraphic重构和次梯度投影算法。其次，提出了一种割面共识方法，用于定位DRCP的近似最优共识解并保证可行性。该方法基于迭代逼近DRCP，通过逐步减小右侧约束的限制参数，并将割面填充到现有的有限约束集中。第三，为确保分布式优化的有限时间终止，基于UJSC图下的共识和零阶停止条件，开发了一种分布式终止算法。第四，证明了割面共识方法可在有限时间内终止，并为每个智能体生成一个可行且近似最优的解。最后，通过数值算例说明了该方法的有效性。