Distributed optimisation approach to least‐squares solution of Sylvester equations

Distributed solver for linear matrix inequalities: an optimization perspective

“…To solve the LMIs with wide applications, the distributed algorithm ( 10) is a typical algorithm, which is consistent with those distributed algorithms of solving matrix equations in [15][16][17]. Except for the original variable x i ∈ ℝ m for every agent i, auxiliary variables i ∈ ℝ and Y i ∈ p are introduced to reformulate the LMIs into a distributed optimization, while dual variables R i ∈ ℝ p×p and i ∈ ℝ m are designed for the equality constraint and the consistency constraint, respectively.…”

“…In [8][9][10][11][12], distributed algorithms were proposed for least squares solutions to Ax = b over a multi-agent network, where each agent only knew one row of A and b. Online distributed algorithm for linear regressions was explored in [13] In [14], distributed algorithms were proposed for least squares solutions of AXB = F , where eight standard data partitions were discussed. The distributed computation for Sylvester equations, Lyapunov equations, Stein equations and algebraic Riccati inequalities was investigated in [15][16][17][18]. To obtain a low-rank solution of a linear matrix equation, Li et al [19] studied the nuclear norm minimization with linear constraints and proposed a distributed primal-dual algorithm.…”