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, an approximated DRCP (ADRCP) is introduced by discretizing the semi-infinite constraints into a finite number of inequality constraints to ensure tractability and restricting the right-hand side of the constraints with a positive parameter to ensure a feasible solution for (DRCP) can be obtained. This problem is iteratively solved by a distributed projected gradient algorithm proposed in this paper, which is based on epigraphic reformulation and gradient projected operations. Secondly, a cutting-surface consensus approach is proposed for locating an approximately optimal consensus solution of the DRCP with guaranteed local feasibility for each agent. This approach is based on iteratively approximating the DRCP by successively reducing the restriction parameter of the right-hand constraints and adding 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：本文针对时变非平衡有向网络在一致联合强连通（UJSC）假设下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以保证可处理性，并利用正参数限制约束右侧以确保能获得（DRCP）的可行解，引入了近似DRCP（ADRCP）。该问题通过本文提出的基于外延重构和梯度投影操作的分布式投影梯度算法进行迭代求解。其次，提出了一种切面共识方法，用于定位DRCP的近似最优共识解，并保证每个智能体的局部可行性。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切面添加到现有有限约束集中来实现。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，证明了切面共识方法能在有限时间内终止，并为每个智能体生成一个可行且近似最优的解。最后，通过数值算例验证了该方法的有效性。