We study dual-based algorithms for distributed convex optimization problems over networks, where the objective is to minimize a sum $\sum _{i=1}^m{f}_i\left(z\right)$ of functions over in a network. We provide complexity bounds for four different cases, namely: each function $f_i$ is strongly convex and smooth, each function is either strongly convex or smooth, and when it is convex but neither strongly convex nor smooth. Our approach is based on the dual of an appropriately formulated primal problem, which includes a graph that models the communication restrictions. We propose distributed algorithms that achieve the same optimal rates as their centralized counterparts (up to constant and logarithmic factors), with an additional optimal cost related to the spectral properties of the network. Initially, we focus on functions for which we can explicitly minimize its Legendre–Fenchel conjugate, i.e. admissible or dual friendly functions. Then, we study distributed optimization algorithms for non-dual friendly functions, as well as a method to improve the dependency on the parameters of the functions involved. Numerical analysis of the proposed algorithms is also provided.

我们研究基于双算法的网络上分布式凸优化问题，其目的是使总和最小化 $\sum _{\mathrm{一世}=\mathrm{1个}}^{米}{F}_{\mathrm{一世}}（ž）$网络中的功能集。我们提供了四种不同情况的复杂度界限，即：每个函数$F_{\mathrm{一世}}$是强凸且平滑的，每个函数要么是强凸或平滑的，当它是凸的但既不是强凸也不是平滑的时。我们的方法基于适当公式化的原始问题的对偶，其中包括对通信限制进行建模的图形。我们提出了一种分布式算法，该算法可实现与集中式算法相同的最佳速率（不超过常数和对数因子），并且具有与网络频谱特性相关的其他最佳成本。最初，我们专注于可以显式最小化其Legendre-Fenchel共轭的函数，即可允许的或双重友好的函数。然后，我们研究了非对偶友好函数的分布式优化算法，以及一种改善对所涉及函数参数的依赖性的方法。