Convergence analysis of the augmented Lagrangian method for $$\ell _{p}$$ -norm cone optimization problems with $$p \ge 2$$

Abstract

This paper focuses on the convergence analysis of the augmented Lagrangian method (ALM) for \(\varvec{\ell }_{\varvec{p}}\)-norm cone optimization problems. We investigate some properties of the augmented Lagrangian function and \(\varvec{\ell }_{\varvec{p}}\)-norm cone. Moreover, under the Jacobian uniqueness conditions, we prove that the local convergence rate of ALM for solving \(\varvec{\ell }_{\varvec{p}}\)-norm cone optimization problems with \(\varvec{p} \varvec{\ge } \varvec{2}\) is proportional to \(\varvec{1}\varvec{/}\varvec{r}\), where the penalty parameter \(\varvec{r}\) is not less than a threshold \(\varvec{\hat{r}}\). In numerical simulations, we successfully validate the effectiveness and convergence properties of ALM.