Since the basis pursuit problem is of the type: $\min_{\alpha} \parallel \alpha\parallel s.t\ y=X\alpha$. Shouldn't we be able to solve it using the Lagrange multiplier? Why is that an option that I never see in documentations?
Thanks !
Edit : https://www.hindawi.com/journals/mpe/2022/8711843/#EEq5 gives me the following solution $diag(|\alpha|) X^T (A\ diag(|\alpha|)\ A^T)^{-1}y$ where $diag(u)$ is a diagonal matrix whose diagonal is taken from vector u. Is that something accurate ?