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Semidefinite programming duality and linear system theory: connections and implications for computation

Several important problems in control theory can be reformulated as semidefinite programming problems (SDPs), i.e., as convex optimization problems with linear matrix inequality (LMI) constraints. From duality theory in convex optimization, dual problems can be derived for these SDPs. These dual problems can in turn be reinterpreted in control or system theoretic terms, often yielding new results or new proofs for existing results from control theory. We explore such connections for a few problems associated with linear time-invariant systems. Specifically, we discuss the following three applications of SDP duality. Theorems of alternatives provide systematic and unified proofs of necessary and sufficient conditions for solvability of LMIs. As an example, we present a simple new proof of the KYP lemma. The dual problem associated with an SDP can be used to derive lower bounds on the optimal value. As an example, we give a duality-based proof of the Enns-Glover lower bound. The optimal solution of an SDP is characterized by necessary and sufficient optimality conditions that involve the dual variables. As an example, we show that the properties of the solution of the LQR problem can be derived directly from the SDP optimality conditions. Several of the results that we use from convex duality require technical conditions (so-called constraint qualifications). We show that for problems involving Riccati inequalities these constraint qualifications are related to controllability and observability. In particular, this leads us to a new criterion for controllability. We also point out some implications of these results for computational methods for large-scale SDPs arising in control.