Convergence Of Augmented Lagrangian Methods For Piecewise Linear Quadratic Composite Optimization Problems

In this paper,the convergence of the inexact augmented Lagrangian method is discussed by taking convex piecewise linear quadratic composite optimization problems as the research object.Firstly,we use the augmented Lagrangian function to characterize the KKT system and the corresponding Lagrange multiplier.And show the equivalence between the noncritical of Lagrange multiplies,Semi-isolated calmness of KKT system solution mapping and Error Bounds of KKT system,all of which can be implied by the second-order sufficient condition.Then,based on the first-order optimality condition of the KKT system,some equivalent conditions are obtained and the framework of the inexact augmented Lagrangian algorithm is given by using the second-order epi-derivative of the augmented Lagrangian.Under the assumption of second-order sufficient conditions,the convergence and convergence rate of the algorithm are proved.When the penalty parameter is large enough,the convergence rate of the primal-dual sequence Q-linear is proved.When the penalty parameter tends to infinity,it becomes super-linear.In the last part,the above results are used to solve the l2 penalty problem of linear quadratic optimization in stochastic optimization.