This paper concerns structured $\ell_0$-norms regularization problems, with a twice continuously differentiable loss function and a box constraint. This class of problems have a wide range of applications in statistics, machine learning and image processing. To the best of our knowledge, there is no efficient algorithm in the literature for solving them. In this paper, we first provide a polynomial-time algorithm to find a point in the proximal mapping of the fused $\ell_0$-norms with a box constraint based on dynamic programming principle. We then propose a hybrid algorithm of proximal gradient method and inexact projected regularized Newton method to solve structured $\ell_0$-norms regularization problems. The iterate sequence generated by the algorithm is shown to be convergent by virtue of a non-degeneracy condition, a curvature condition and a Kurdyka-Łojasiewicz property. A superlinear convergence rate of the iterates is established under a locally Hölderian error bound condition on a second-order stationary point set, without requiring the local optimality of the limit point. Finally, numerical experiments are conducted to highlight the features of our considered model, and the superiority of our proposed algorithm.

For (nearly) sparse signal reconstruction problems, we propose an inexact sequential convex relaxation algorithm (iSCRA-TL1) by constructing the working index set iteratively with a simple and adaptive strategy, and solving inexactly a sequence of truncated$\ell _{1}$-norm minimization subproblems. A toy example is provided to demonstrate that the exact version of iSCRA-TL1 can successfully reconstruct the true sparse signal, but almost all the present sequential convex relaxation algorithms starting from an optimal solution of the$\ell _{1}$-norm minimization fail to recover it. To provide theoretical guarantees for iSCRA-TL1, we introduce two new types of null space properties, restricted null space property (RNSP) and sequential restricted null space property (SRNSP), and prove that they are both weaker than the common stable NSP, while their robust versions are not stronger than the existing robust NSP. Then, we justify that under a suitable (robust) SRNSP, iSCRA-TL1 can identify the support of the true r-sparse signal or the index set of the first r largest (in modulus) entries of the true nearly r-sparse signal via at most r truncated$\ell _{1}$-norm minimization, and the error bound of its final output from the true (nearly) r-sparse signal is also quantified. To the best of our knowledge, this is the first sequential convex relaxation algorithm to recover the support of the true (nearly) sparse signal under a weaker NSP condition within a specific number of steps, provided that the classical$\ell _{1}$-norm minimization problem lacks the good robustness.