Correction to: On the best achievable quality of limit points of augmented Lagrangian schemes

The published article “R. Andreani, G. Haeser, L. M. Mito, A. Ramos, L. D. Secchin. On the best achievable quality of limit points of augmented Lagrangian schemes. Numer. Algor., 2021. https://doi.org/10.1007/s11075-021-01212-8” has the errors listed below. The authors apologize for that.

• Step 1, Algorithms 2 and 3: $\|\tilde V^{i_{k}-\frac {1}{2}}\|_{\infty }\leqslant \varepsilon _{k}$ should be $\| \nabla L_{\rho _{k},\bar \lambda ^{k},\bar \mu ^{k}}(x^{k})\|_{\infty }\leqslant \varepsilon _{k}$

• Expression (11): $\frac {\|x^{k}-x^{*}\|_{2}}{\rho _{k}}\to 0$ should be $\frac {\| (\bar \lambda ^{k} , \bar \mu ^{k}) \|_{2}}{\rho _{k}}\to 0$

• Paragraph after Theorem 5: $\|\bar \lambda ^{k} , \bar \mu ^{k}\|_{2} = O(\rho _{k}^{\beta })$, should be $\| (\bar \lambda ^{k} , \bar \mu ^{k}) \|_{2} = O(\rho _{k}^{\beta })$ and ∥x^k − x^∗∥₂ = O(ρ_k) should be $\| (\bar \lambda ^{k} , \bar \mu ^{k}) \|_{2} = O(\rho _{k})$

• Expression (16): $\|\tilde V^{i_{k}-\frac {1}{2}}\|_{\infty }\to 0$ should be $\| \text {proj}_{{{\varOmega }}(x^{k})} (x^{k} - \nabla f(x^{k})) - x^{k} \|_{\infty }\to 0$

• Definition of K^P(x,α,β), after Theorem 7: ∥x^k − x^∗∥₂ should be $\| (1,\lambda ,\mu ) \|_{\infty }$

• Item 2. after Definition 1: $\{ \lambda ^{k}, \mu ^{k} \}_{k\in \mathbb N}$ should be $\{ (\lambda ^{k}, \mu ^{k} ) \}_{k\in \mathbb N}$

• Example 1 and step 2 of Algorithm 3: $\bar {{\mu ^{k}_{1}}}, \bar {{\mu ^{k}_{2}}}$ and $\bar {{\mu ^{k}_{p}}}$ should be ${\bar \mu ^{k}_{1}}, {\bar \mu ^{k}_{2}}$ and ${\bar \mu ^{k}_{p}}$, respectively