Gossip Algorithms for Convex Consensus Optimization Over Networks

Abstract

In many applications, nodes in a network desire not only a consensus, but an optimal one. To date, a family of subgradient algorithms have been proposed to solve this problem under general convexity assumptions. This paper shows that, for the scalar case and by assuming a bit more, novel non-gradient-based algorithms with appealing features can be constructed. Specifically, we develop Pairwise Equalizing (PE) and Pairwise Bisectioning (PB), two gossip algorithms that solve unconstrained, separable, convex consensus optimization problems over undirected networks with time-varying topologies, where each local function is strictly convex, continuously differentiable, and has a minimizer. We show that PE and PB are easy to implement, bypass limitations of the subgradient algorithms, and produce switched, nonlinear, networked dynamical systems that admit a common Lyapunov function and asymptotically converge. Moreover, PE generalizes the well-known Pairwise Averaging and Randomized Gossip Algorithm, while PB relaxes a requirement of PE, allowing nodes to never share their local functions.

Distributed Subgradient Algorithm for Multi-Agent Convex Optimization with Global Inequality and Equality Constraints

“…Distributed optimization problems of multi-agent systems appear different kinds of distributed processing issues such as distributed estimation, distributed motion planning, distributed resource allocation and distributed congestion control [1][2][3][4][5][6][7][8][9][10][11][12]. The main focus is to solve a distributed optimization problem where the global objective function is composed of a sum of local objective functions, each of which is only known by one agent.…”

Consensus Achievement of Decentralized Sensors Using Adapted Particle Swarm Optimization Algorithm

“…In [13], the goal of gossiping is described as for the agents to reach a consensus in the sense that all gossip variables ultimately reach the same value at the average of the initial values: x in the limit as → ∞. As an application of the algorithm, Lu et al [14] proposed a pairwise equalizing as a gossip-style distributed asynchronous iterative algorithm for achieving convex consensus optimization over undirected networks.…”

Inexact dual averaging method for distributed multi-agent optimization

“…Average consensus algorithms [3][4][5][6][7][8][9][10][11][12], where each agent in the network has to agree on a common variable via local information exchange, have been utilized to develop the distributed methods [2,[13][14][15][16][17][18][19][20][21][22][23]. Among them, the authors in [13] propose the distributed subgradient method, which consists of two steps, namely, a consensus step and a subgradient step; each agent takes the subgradient step to minimize its own objective function, while the consensus step is taken by each agent to coordinate its estimate with its neighbors.…”