In the context of linear constrained optimization, we study in this paper the problem of finding an optimal solution satisfying all but k of the given n constraints. A solution is obtained by means of an algorithm of the complexity min{O(n.k(d)), O(n.d(k+1))}, where d is the dimension of the problem. We then use these results to solve the problem of robust identification in the presence of outliers in the setting of bounded error parameter identification. Finally, we show that the estimate obtained converges to the true but unknown parameter in the presence of outliers.