We consider distributed convex optimization problems that involve a separable objective function and nontrivial functional constraints, such as linear matrix inequalities. We propose a decentralized and computationally inexpensive algorithm, which is based on the concept of approximate projections. Our algorithm is one of the consensus-based methods in that, at every iteration, each agent performs a consensus update of its decision variables followed by an optimization step of its local objective function and local constraints. Unlike other methods, the last step of our method is not a Euclidean projection onto the feasible set, but instead a subgradient step in the direction that minimizes the local constraint violation. We propose two different averaging schemes to mitigate the disagreements among the agents’ local estimates. We show that the algorithms converge almost surely, i.e., every agent agrees on the same optimal solution, under the assumption that the objective functions and constraint functions are nondifferentiable and their subgradients are bounded. We provide simulation results on a decentralized optimal gossip averaging problem, which involves semidefinite programming constraints, to complement our theoretical results.

中文翻译：

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具有功能约束的分散优化的近似投影方法


我们考虑涉及可分离目标函数和非平凡函数约束（例如线性矩阵不等式）的分布式凸优化问题。我们提出了一种分散且计算成本低廉的算法，该算法基于近似投影的概念。我们的算法是基于共识的方法之一，因为在每次迭代时，每个代理都会对其决策变量执行共识更新，然后对其局部目标函数和局部约束进行优化步骤。与其他方法不同，我们方法的最后一步不是对可行集的欧几里德投影，而是朝着最小化局部约束违规的方向的次梯度步骤。我们提出了两种不同的平均方案来减轻代理人本地估计之间的分歧。我们证明，在目标函数和约束函数不可微且其次梯度有界的假设下，算法几乎肯定会收敛，即每个代理都同意相同的最优解。我们提供了分散式最优八卦平均问题的模拟结果，其中涉及半定规划约束，以补充我们的理论结果。