Abstract
In this paper, we study the minimization of l p-q (0 <; p ≤ 1, q ≥ 1, p=6q), 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 ≥ 1) and e-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.
Highlights
Numerous optimization models and techniques, such as compressed sensing (CS) [1]–[3] and phaseless compressed sensing (PCS) [4]–[6], have been proposed for finding sparse solutions to a system or an optimization problem
For the sensing matrices satisfying a variant of the restricted isometry property (RIP), we found that the ε-approximation-based optimization method is better than the difference of convex functions algorithm (DCA)-based optimization method for improving the sparsity of solutions
We propose IRLt,p−q for lp−q regularized unconstrained minimization based on the iterative reweighted minimization method
Summary
Numerous optimization models and techniques, such as compressed sensing (CS) [1]–[3] and phaseless compressed sensing (PCS) [4]–[6], have been proposed for finding sparse solutions to a system or an optimization problem. We will consider the influences of the parameters p and q and investigate the sparsity of minimizers and recovery conditions referring to this generic optimization Based on these key facts, we further propose iteratively reweighted minimization algorithms to solve (5) and focus on the convergence of algorithms. ITERATIVE REWEIGHTED lp−q -MINIMIZATION In general, the principle of the IRLS method for nonconvex optimization is that, as the nonconvex part in the optimization problem does not change, it may be possible to find a single convex function that, weighted appropriately, can serve as majorizer for the nonconvex at each step of the proposed algorithms [16]–[18], [24].
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