An Accelerated Proximal Alternating Direction Method of Multipliers for Optimal Decentralized Control of Uncertain Systems

Abstract

To ensure the system stability of the ${\mathcal{H}}_2$-guaranteed cost optimal decentralized control (ODC) problem, we formulate an approximate semidefinite programming (SDP) problem that leverages the block diagonal structure of the decentralized controller’s gain matrix. To minimize data storage requirements and enhance computational efficiency, we employ the Kronecker product to vectorize the SDP problem into a conic programming (CP) problem. We then propose a proximal alternating direction method of multipliers (PADMM) to solve the dual of the resulting CP problem. By using the equivalence between the semi-proximal ADMM and the (partial) proximal point algorithm, we identify the non-expansive operator of PADMM, enabling the use of Halpern fixed-point iteration to accelerate the algorithm. Finally, we demonstrate that the sequence generated by the proposed accelerated PADMM exhibits a fast convergence rate for the Karush–Kuhn–Tucker residual. Numerical experiments confirm that the accelerated algorithm outperforms the well-known COSMO, MOSEK, and SCS solvers in efficiently solving large-scale CP problems, particularly those arising from ${\mathcal{H}}_2$-guaranteed cost ODC problems.