Summary In compressive sampling theory, the least absolute shrinkage and selection operator (LASSO) is a representative problem. Nevertheless, the non‐differentiable constraint impedes the use of Lagrange programming neural networks (LPNNs). We present in this article the 𝒫k‐LPNN model, a novel algorithm that tackles the LASSO minimization together with the underlying theory support. First, we design a sequence of smooth constrained optimization problems, by introducing a convenient differentiable approximation to the non‐differentiable l1‐norm constraint. Next, we prove that the optimal solutions of the regularized intermediate problems converge to the optimal sparse signal for the LASSO. Then, for every regularized problem from the sequence, the 𝒫k‐LPNN dynamic model is derived, and the asymptotic stability of its equilibrium state is established as well. Finally, numerical simulations are carried out to compare the performance of the proposed 𝒫k‐LPNN algorithm with both the LASSO‐LPNN model and a standard digital method.