Quasi-Newton L-BFGS based inexact primal–dual proximal point algorithms to solve nonsmooth convex composite programs for sparse signal recovery applications

In this work, we first propose a new inexact primal–dual proximal point algorithm (iPDPPA) for solving a general nonsmooth convex composite model with three focuses on the basis pursuit, the quadratically constrained ${\ell }_1$-minimization and the Dantzig selector in the context of compressive sensing. We then prove its global convergence and discuss its convergence rate in some cases. We also prove that the objective function in the first subproblem of the proposed scheme is strongly convex and continuously differentiable, and then apply the quasi-Newton L-BFGS method to solve the subproblem. Numerical experiments show the effectiveness of L-BFGS-iPDPPA on sparse signal recovery.