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We introduce a new family of algorithms to exploit sparsity in adaptive filters. It is based on a recently introduced new framework for designing robust adaptive filters. It results from minimizing a certain cost function subject to a time-dependent constraint on the norm of the filter update. Although in general this problem does not have a closed-form solution, we propose an approximate one which is very close to the optimal solution. We take a particular algorithm from this family and provide some theoretical results regarding the asymptotic behavior of the algorithm. Finally, we test it in different environments for system identification and acoustic echo cancellation applications.