%0 Journal Article
%@ 0005-1098
%A Piga, Dario
%A Tóth, Roland
%D 2013
%F eprints:2474
%I Elsevier 
%J Automatica
%K Compressive sensing; ℓ0ℓ0-minimization; Regularization; SDP relaxation; Sparse estimation; Segmentation
%N 12
%P 3646 - 3653
%T An SDP approach for l0-minimization: application to ARX model segmentation 
%U http://eprints.imtlucca.it/2474/
%V 49
%X Abstract Minimizing the ℓ 0 -seminorm of a vector under convex constraints is a combinatorial (NP-hard) problem. Replacement of the ℓ 0 -seminorm with the ℓ 1 -norm is a commonly used approach to compute an approximate solution of the original ℓ 0 -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 ℓ 1 -approximated problem is equal to the solution of the original ℓ 0 -minimization problem. However, the evaluation of the conservativeness of the ℓ 1 -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 ℓ 0 -norm of a vector under given constraints. In particular, we show that an ℓ 0 -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 ℓ 1 -based relaxation in detecting piece-wise constant parameter changes in the estimated model.