Dual-density-based reweighted $$\ell _{1}$$-algorithms for a class of $$\ell _{0}$$-minimization problems

The optimization problem with sparsity arises in many areas of science and engineering such as compressed sensing, image processing, statistical learning and data sparse approximation. In this paper, we study the dual-density-based reweighted $$\ell _{1}$$ ℓ 1 -algorithms for a class of $$\ell _{0}$$ ℓ 0 -minimization models which can be used to model a wide range of practical problems. This class of algorithms is based on certain convex relaxations of the reformulation of the underlying $$\ell _{0}$$ ℓ 0 -minimization model. Such a reformulation is a special bilevel optimization problem which, in theory, is equivalent to the underlying $$\ell _{0}$$ ℓ 0 -minimization problem under the assumption of strict complementarity. Some basic properties of these algorithms are discussed, and numerical experiments have been carried out to demonstrate the efficiency of the proposed algorithms. Comparison of numerical performances of the proposed methods and the classic reweighted $$\ell _1$$ ℓ 1 -algorithms has also been made in this paper.

The Rate of Convergence of a NLM Based on F–B NCP for Constrained Optimization Problems Without Strict Complementarity

It is well-known that the linear rate of convergence can be established for the classical augmented Lagrangian method for constrained optimization problems without strict complementarity. Whether this result is still valid for other nonlinear Lagrangian methods (NLM) is an interesting problem. This paper proposes a nonlinear Lagrangian function based on Fischer–Burmeister (F–B) nonlinear complimentarity problem (NCP) function for constrained optimization problems. The rate of convergence of this NLM is analyzed under the linear independent constraint qualification and the strong second-order sufficient condition without strict complementarity when subproblems are assumed to be solved exactly and inexactly, respectively. Interestingly, it is demonstrated that the Lagrange multipliers associating with inactive inequality constraints at the local minimum point converge to zeros superlinearly. Several illustrative examples are reported to show the behavior of the NLM.

AN INFEASIBLE SSLE FILTER ALGORITHM FOR GENERAL CONSTRAINED OPTIMIZATION WITHOUT STRICT COMPLEMENTARITY

In this paper, we propose a new sequential systems of linear equations (SSLE) filter algorithm, which is an infeasible QP-free method. The new algorithm needs to solve a few reduced systems of linear equations with the same nonsingular coefficient matrix, and after finitely many iterations, only two linear systems need to be solved. Furthermore, the nearly active set technique is used to improve the computational effect. Under the linear independence condition, the global convergence is proved. In particular, the rate of convergence is proved to be one-step superlinear without assuming the strict complementarity condition. Numerical results and comparison with other algorithms indicate that the new algorithm is promising.