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

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