Convergence Of Inexact Augmented Lagrangian Methods For Optimization Problems With Sparse Constraint

Sparse optimization plays an important role in machine learning,pattern recognition,compressed sensing,noise processing,computer image processing and other aspects.It is a hot topic in the research of many experts and scholars.In recent years,many scholars are interested in the optimality conditions of sparse optimization problems.In this paper,we study sparsity optimization problem with the set inclusion constraints.For inexact augmented Lagrangain method,the convergence and convergence rate is discussed.Firstly,the KKT system is established by using the Lagrangian multiplier subset is defined.According to the first-order optimality condition of the KKT system,some equivalent conditions are obtained and the framework of inexact augmented Lagrangian algorithm is given.Then,it is proved that the semi-isolated calm property of solution mapping of the generalized equivalent which is composed of the perturbed KKT system is equivalent to the error bound of KKT system.The locally uniform second-order growth conditions of augmented Lagrangian functions are described by second-order sufficient conditions.Finally,under the assumptions of second order sufficient conditions and semi-isolated calm,the -linear convergence rate for the sequence generated by algorithm is proved.When the penalty parameter tends to infinity,it becomes superlinear convergence.