The reason is because adding the L 1 norm will make the objective function not continuously differentiable, where as the L 1 constraint could be just a simple linear constraint (Lee et al., 2006).
the alternative L 1 -constrained problem could be defined as: To test the robustness of Laplacian structured sparsity term in the setup of a L 1 -constrained problem, we can incorporate the Laplacian penalty term into the above formula, and derive: Note that the Laplacian matrix is positivesemidefinite, because this graph Laplacian penalty can be viewed as a quadratic term, and the objective function in equation 19 is now convex differentiable and will produce sparse estimates, so that we are able to use a limited-memory projected quasi-Newton method (Schmidt et al., 2009) to solve the dual form of this problem.