An augmented Lagrangian method for a class of LMI-constrained problems in robust control theory

This paper present a new approach to solve a class of nonconvex LMI-constrained problem in robust control theory. The problems we consider may be recast as the minimization of a linear objective subject to linear matrix inequality (LMI) constraints in tandem with nonconvex constraints related to rank conditions. We solve these problems using an extension of the augmented Lagrangian technique. The Lagrangian function combines a multiplier term and a penalty term governing the nonconvex constraints. The LMI constraints, due to their special structure, are handled explicitly and not included in the Lagrangian. Global and fast local convergence of our approach is then obtained by LMI-constrained Newton type method including line search strategy. This procedure may therefore be regarded as a sequential semi-definite programming (SSDP) method, inspired by the sequential quadratic programming (SQP) in nonlinear optimization. The method is conveniently implemented with available SDP interior-point solvers. We compare its performance to the well-known D-K iteration scheme in robust control. Two test problems are investigated and demonstrate the power and efficiency of our approach