  In compressed sensing, the sensing matrix is assumed perfectly known.
However, there exists perturbation in the sensing matrix in reality due to
sensor offsets or noise disturbance. Directions-of-arrival (DoA) estimation
with off-grid effect satisfies this situation, and can be formulated into a
(non)convex optimization problem with linear inequalities constraints, which
can be solved by the interior point method (using the CVX tools), but at a
large computational cost. In this work, in order to design efficient
algorithms, we consider various alternative formulations, such as unconstrained
formulation, primal-dual formulation, or conic formulation to develop
group-sparsity promoted solvers. First, the consensus alternating direction
method of multipliers (C-ADMM) is applied. Then, iterative algorithms for the
BPDN formulation is proposed by combining the Nesterov smoothing technique with
accelerated proximal gradient method, and the convergence analysis of the
method is conducted as well.
  We also developed a variant of EGT (Excessive Gap Technique)-based
primal-dual method to systematically reduce the smoothing parameter
sequentially. Finally, we propose algorithms for quadratically constrained
L2-L1 mixed norm minimization problem by using the smoothed dual conic
optimization (SDCO) and continuation technique. The performance of accuracy and
convergence for all the proposed methods are demonstrated in the numerical
simulations.
