  We propose a new algorithmic framework for constrained compressed sensing
models that admit nonconvex sparsity-inducing regularizers including the
log-penalty function as objectives, and nonconvex loss functions such as the
Cauchy loss function and the Tukey biweight loss function in the constraint.
Our framework employs iteratively reweighted $\ell_1$ and $\ell_2$ schemes to
construct subproblems that can be efficiently solved by well-developed solvers
for basis pursuit denoising such as SPGL1 [6]. We propose a new termination
criterion for the subproblem solvers that allows them to return an infeasible
solution, with a suitably constructed feasible point satisfying a descent
condition. The feasible point construction step is the key for establishing the
well-definedness of our proposed algorithm, and we also prove that any
accumulation point of this sequence of feasible points is a stationary point of
the constrained compressed sensing model, under suitable assumptions. Finally,
we compare numerically our algorithm (with subproblems solved by SPGL1 or the
alternating direction method of multipliers) against the SCP$_{\rm ls}$ in [41]
on solving constrained compressed sensing models with the log-penalty function
as the objective and the Cauchy loss function in the constraint, for
badly-scaled measurement matrices. Our computational results show that our
approaches return solutions with better recovery errors, and are always faster.
