Piga, Dario and Tóth, Roland
An SDP approach for l0minimization: application to ARX model segmentation.
Automatica, 49 (12).
3646  3653.
ISSN 00051098
(2013)
Abstract
Abstract Minimizing the ℓ 0 seminorm of a vector under convex constraints is a combinatorial (NPhard) 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 piecewise constant parameter changes in the estimated model.
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