Log-Barrier Search for Structural Linear Quadratic Regulators

This article studies the design of linear quadratic regulators (LQR) subject to structural constraints, which remains an NP-hard open problem. Both state-feedback and static output-feedback cases are investigated. Instead of using case-by-case relaxation techniques, we propose a tractable first-order method to solve this structural optimal control problem. More specifically, we equivalently reformulate it as a constrained optimization and characterize its first-order optimality condition via Karush–Kuhn–Tucker conditions. To solve this NP-hard problem, we propose a novel optimization method, called log-barrier search (LBS), which incorporates a modified log-barrier term into the control objective function and adaptively changes its parameters during the computation process. The convergence of our method is theoretically guaranteed and at least a stationary solution can be obtained. We compare the proposed method with other existing algorithms, where the LBS method shows a very competitive performance in both speed and optimality.