Iterative Reweighted Minimization for Generalized Norm/Quasi-Norm Difference Regularized Unconstrained Nonlinear Programming

In this paper, we study the minimization of
$l_{p-q}$
(
$0 < p\le 1,~q\ge 1,~p \ne q$
), the general difference of
$l_{p}$
and
$l_{q}$
norms/quasi-norms, as a nonconvex metric for solving unconstrained nonlinear programming. We first establish an exact (stable) sparse recovery condition for the
$l_{p-q}$
constrained problem under an extended restricted
$p$
-isometry property and then propose an iterative algorithm for the
$l_{p-q}$
regularized unconstrained minimization based on the
$t$
-variant of the iterative reweighted minimization method (
$t\ge 1$
) and
$\varepsilon $
-approximation. We theoretically prove that the proposed algorithm converges to a stationary point satisfying the first-order optimality condition. In particular, we provide a convergence rate analysis of the method and show that the local convergence is superlinear under a certain condition. Our extensive experimental results demonstrate that if the sensing matrix satisfies the restricted
$p$
-isometry property, the proposed iterative reweighted minimization method for the
$l_{p-q}$
unconstrained problem generally outperforms the existing methods (especially for those methods based on the difference of norms). For the ill-conditioned sensing matrix, a variant of our method via the difference of convex functions algorithm (DCA) shows better performance on the frequency of success for signal sparse recovery. Likewise, our methods are illustrated to be valid and generally outperform the existing methods for real images.