An SDP approach for ℓ0ℓ0-minimization: Application to ARX model segmentation

Minimizing the ℓ0ℓ0-seminorm of a vector under convex constraints is a combinatorial (NP-hard) problem. Replacement of the ℓ0ℓ0-seminorm with the ℓ1ℓ1-norm is a commonly used approach to compute an approximate solution of the original ℓ0ℓ0-minimization problem by means of convex programming. In the theory of compressive sensing, the condition that the sensing matrix satisfies the Restricted Isometry Property (RIP) is a sufficient condition to guarantee that the solution of the ℓ1ℓ1-approximated problem is equal to the solution of the original ℓ0ℓ0-minimization problem. However, the evaluation of the conservativeness of the ℓ1ℓ1-relaxation approaches is recognized to be a difficult task in case the RIP is not satisfied. In this paper, we present an alternative approach to minimize the ℓ0ℓ0-norm of a vector under given constraints. In particular, we show that an ℓ0ℓ0-minimization problem can be relaxed into a sequence of semidefinite programming problems, whose solutions are guaranteed to converge to the optimizer (if unique) of the original combinatorial problem also in case the RIP is not satisfied. Segmentation of ARX models is then discussed in order to show, through a relevant problem in system identification, that the proposed approach outperforms the ℓ1ℓ1-based relaxation in detecting piece-wise constant parameter changes in the estimated model.