This paper performs further improvement to a distributed algorithm for solving linear algebraic equations via multi-agent networks recently developed by Mou et al., in which all agents' states converge exponentially fast to the same solution to a group of linear equations by assuming each agent knows only part of the linear equations and its nearby neighbors' states. We first prove that the algorithm proposed by Mou et al. with special initialization is able to achieve the solution that is closest to a given point in the Euclidean distance. Second, we eliminate the required initialization step used by Mou et al. by a modification to the update equation. Both analytical and numerical results are provided for validation.

This paper proposes scalable, distributed algorithms for solving linear equations by integrating two mechanisms, termed consensus and conservation, in double-layered multiagent networks. The multi-agent network considered in the paper is composed of clusters and each cluster consists of an aggregator and a sub-network of agents. By achieving consensus and conservation through agent-agent communications in the same cluster and aggregator-aggregator communications among different clusters, respectively, distributed algorithms are devised for agents to cooperatively achieve a solution to the overall linear equation. These algorithms outperform existing algorithms, including but not limited to the following aspects: first, each agent does not have to know as much as a complete row or column of the overall equation; second, each agent only needs to control as few as two scalar states when the number of clusters and the number of agents are sufficiently large; third, the dimensions of agents' states in the proposed algorithms do not have to be the same (while in contrast, algorithms based on the idea of standard consensus inherently require all agents' states to be of the same dimension)

Consider a set of vectors in $ \mathbb{R}^n $, partitioned into two classes: normal vectors and malicious vectors, for which the number of malicious vectors is bounded but their identities are unknown. The paper provides an efficient way for achieving a resilient convex combination, which is a convex combination of only normal vectors. Compared with existing approaches based on Tverberg points, the proposed method based on the intersection of convex hulls has lower computational complexity. Simulations suggest that the proposed method can be applied to achieve resilience of consensus-based distributed algorithms against Byzantine attacks based only on agents' locally available information.

Inspired and underpinned by the idea of integral feedback, a distributed constant gain algorithm is proposed for multi-agent networks to solve convex optimization problems with local linear constraints. Assuming agent interactions are modeled by an undirected graph, the algorithm is capable of achieving the optimum solution with an exponential convergence rate. Furthermore, inherited from the beneficial integral feedback, the proposed algorithm has attractive requirements on communication bandwidth and good robustness against disturbance. Both analytical proof and numerical simulations are provided to validate the effectiveness of the proposed distributed algorithms in solving constrained optimization problems.

This paper proposes a continuous-time distributed algorithm for multi-agent networks to achieve a solution with the minimum $l_1$ -norm to under-determined linear equations. The proposed algorithm comes from a combination of the Filippov set-valued map with the projection-consensus flow. Given the underlying network is undirected and fixed, it is shown that the proposed algorithm drives all agents' individual states to converge in finite time to a common value, which is the minimum $l_1$ -norm solution.