A second-order multi-agent distributed optimisation algorithm on matrix-weighted networks

This article investigates a second-order multi-agent distributed optimisation algorithm on matrix-weighted networks. To address the limitations of traditional methods on matrix-weighted networks, a novel second-order distributed optimisation algorithm is proposed. By constructing auxiliary variables, the algorithm eliminates the requirement for velocity information updates. Under the proposed control protocol, algebraic and graph-theoretic conditions for achieving optimisation consensus are established based on matrix theory and Barbalat's lemma. Specifically, the given parameter conditions and local cost functions are convex; the second-order consensus at the global optimal solution is achieved when the Laplacian matrix's null space spans the consensus subspace or the matrix-weighted graph contains a positive spanning tree. Finally, a numerical experiment is conducted to validate the effectiveness of proposed algorithms.