We consider a generic decentralized constrained optimization problem over static, directed communication networks, where each agent has exclusive access to only one convex, differentiable, local objective term and one convex constraint set. For this setup, we propose a novel decentralized algorithm, called DAGP (Double Averaging and Gradient Projection), based on local gradients, projection onto local constraints, and local averaging. We achieve global optimality through a novel distributed tracking technique we call distributed null projection. Further, we show that DAGP can be used to solve unconstrained problems with non-differentiable objective terms with a problem reduction scheme. Assuming only smoothness of the objective terms, we study the convergence of DAGP and establish sub-linear rates of convergence in terms of feasibility, consensus, and optimality, with no extra assumption (e.g. strong convexity). For the analysis, we forego the difficulties of selecting Lyapunov functions by proposing a new methodology of convergence analysis in optimization problems, which we refer to as aggregate lower-bounding. To demonstrate the generality of this method, we also provide an alternative convergence proof for the standard gradient descent algorithm with smooth functions. Finally, we present numerical results demonstrating the effectiveness of our proposed method in both constrained and unconstrained problems. In particular, we propose a distributed scheme by DAGP for the optimal transport problem with superior performance and speed.

翻译：本文考虑静态有向通信网络上的通用分布式约束优化问题，其中每个智能体仅能访问自身凸可微的局部目标函数项及一个凸约束集。为此，我们提出一种基于局部梯度、局部约束投影和局部平均的新型分布式算法——DAGP（双重平均与梯度投影）。通过一种称为分布式零投影的新型分布式追踪技术，我们实现了全局最优性。进一步证明，借助问题归约方案，DAGP可应用于含不可微目标项的无约束问题求解。在仅假设目标项光滑性的前提下，我们研究了DAGP的收敛性，并在无额外假设（如强凸性）的情况下，建立了关于可行性、一致性和最优性的次线性收敛率。为规避Lyapunov函数选取的困难，我们提出一种名为聚合下界法的优化问题收敛性分析新方法。通过将该方法应用于标准梯度下降算法（光滑函数情形）的收敛性证明，验证了其通用性。最后，通过数值实验结果展示了所提方法在约束与无约束问题中的有效性，特别地，我们利用DAGP提出了一种用于最优传输问题的分布式方案，该方案具有卓越的性能与运算速度。