In this paper, a filter QP-free infeasible method with nonmonotone line search is proposed for minimizing a smooth optimization problem with smooth inequality constraints. This proposed method is based on the solution of nonsmooth equations, which are obtained by the Lagrangian multiplier method and the function of the nonlinear complementarity problem for the Karush–Kuhn–Tucker optimality conditions. Especially, each iteration of this method can be viewed as a perturbation of a Newton or quasi-Newton iteration on both the primal and dual variables for the solution of the Karush–Kuhn–Tucker optimality conditions. What is more, it is considered to use the function of the nonlinear complementarity problem in the filter, which makes the proposed algorithm avoid the incompatibility. Then the global convergence of the proposed method is given. And under some mild conditions, the superlinear convergence rate can be obtained. Finally, some preliminary numerical results are shown to illustrate that the proposed filter QP-free infeasible method is quite promising.

In terms of variable selection and dimensionality reduction for high-dimensional data, Fused-LASSO is a powerful model when the number of important variables is small and neighboring variables exhibit a high degree of correlation. However, solving the model is challenging due to its non-smoothness and non-separability. Alternating direction method of multipliers (ADMM) is an effective method to solve the Fused-LASSO, but one of the subproblems may have no closed-form solution leading to additional computational overhead, and the computing for the inverse of matrix in the subproblem is expensive. In order to overcome these shortcomings and have a great speed performance, an extended linearized alternating direction method of multipliers (ELADMM) is proposed in this paper. First, in the framework of ADMM, we embed the linearized technique into the x-subproblem, so that the subproblem can be solved quickly and efficiently rather than by inner iteration. Second, we make additional extensions to some variables after each iteration, which improves the speed performance of the LADMM. Numerical results show that the proposed ELADMM outperforms ADMM as well as other existing solvers in the sense that it finds the same accurate variable selection results with the shortest time.

It is well known that the nuclear norm minimization problems are widely used in numerous fields such as machine learning, system recognition, and image processing, etc. It has captured considerable attention and taken good progress. Many researchers have made great contributions to the nuclear norm minimization problem with $ l_{2} $ norm fidelity term, which is just able to deal with those problems with Gaussian noise. In this paper, we propose an efficient penalty decomposition(PD) method to solve the nuclear norm minimization problem with $ l_{1} $ norm fidelity term. One subproblem admits a closed-form solution due to its special structure, and another can also get a closed-form solution by linearizing its quadratic term. The convergence results of the proposed method are given as well. Finally, the effectiveness and efficiency of the proposed method are verified by some numerical experiments.

Sparsity is the essential characteristic of high-dimensional big data problems. Fused least absolute shrinkage and selection operator (FLASSO) model is an important tool for dealing with sparse problems in high-dimensional multivariate linear regression analysis. In this paper, a new linearized alternating direction method of multipliers (NLADMM) is proposed to solve the FLASSO. We present the range of multipliers of the adaptive step size and the relax factors, and then present an inexact version of NLADMM (INLADMM) to make the algorithm efficient and implement easily. Convergence analysis of the proposed algorithms is established under some mild conditions. Numerical results for solving FLASSO with fixed penalty factors show that INLADMM and NLADMM have better variable selection capabilities than the existing methods such as ADMM, LADMM and ELADMM. Especially, INLADMM also obtains solutions with higher accuracy in fewer time than ADMM, ELADMM and NLADMM.