1 Corrections to: J Glob Optim https://doi.org/10.1007/s10898-016-0440-6

This note lists corrections to various errors in a recent article [1] by Khan, Watson, and Barton. Though these errors appear in the text of [1], they were not present in the C++ implementation used in Section 7 of [1]; hence, the examples in that section were not affected by these errors.

  • In the bottom row of Table 1 of [1], concerning relaxations of \(\frac{1}{\xi ^{2k-1}}\) for \(k\in \mathbb {N}\), the entry in the leftmost column should be “\(\mathbb {R}_-\)” instead of \(``\mathbb {R}_+\)”. (The \(B:=\mathbb {R}_+\) case is addressed by the earlier row concerning \(\tfrac{1}{\xi ^k}\).)

  • Proposition 6 of [1] concerns the procedure for obtaining \(\mathscr {C}^2\) relaxations of expressions involving odd powers. In this proposition, in the construction of \(\overline{\phi }^{\mathrm {C}}\), the “\(\max \)” function should instead be “\(\min \)”; the corrected construction is:

  • In Definition 3 of [1], in the constructions of \(x^*\) and \(y^*\), the \(\sigma _\mu \) terms should be subtracted rather than added. This affects the Whitney-\(\mathscr {C}^1\) relaxations of products described in Theorem 6. The corrected constructions are:

    $$\begin{aligned} x^*:(y,\varvec{\zeta },\varvec{\eta })&\mapsto \underline{\zeta }+(\overline{\zeta }-\underline{\zeta })\left( \frac{\overline{\eta }-y}{\overline{\eta }-\underline{\eta }} - \sigma _\mu \left( \tfrac{\underline{\eta }+\overline{\eta }}{(\mu +1)(\overline{\eta }-\underline{\eta })}\right) \right) , \\ y^*: (x,\varvec{\zeta },\varvec{\eta })&\mapsto \underline{\eta }+(\overline{\eta }-\underline{\eta })\left( \frac{\overline{\zeta }-x}{\overline{\zeta }-\underline{\zeta }} -\sigma _\mu \left( \tfrac{\underline{\zeta }+\overline{\zeta }}{(\mu +1)(\overline{\zeta }-\underline{\zeta })}\right) \right) . \end{aligned}$$

    The proof of Theorem 6 in [1] is valid after this correction.

  • Proposition 15 of [1] provides partial derivatives for the relaxations of products described in Theorem 6. In Proposition 15, in the provided expressions for partial derivatives of \(\underline{\psi }_{\times ,\mathrm {A}}\), the exponents should be \(\mu -1\) rather than \(\mu +1\). The corrected partial derivatives are:

    $$\begin{aligned} \frac{\partial {\underline{\psi }_{\times ,\mathrm {A}}}}{\partial {x}}(x,y,\varvec{\zeta },\varvec{\eta })&= \frac{1}{2}\left( \underline{\eta }+\overline{\eta } + (\mu +1)(\overline{\eta }-\underline{\eta }) \left( \tfrac{y-\underline{\eta }}{\overline{\eta }-\underline{\eta }} - \tfrac{\overline{\zeta }-x}{\overline{\zeta }-\underline{\zeta }}\right) \left| \tfrac{y-\underline{\eta }}{\overline{\eta }-\underline{\eta }} - \tfrac{\overline{\zeta }-x}{\overline{\zeta }-\underline{\zeta }}\right| ^{\mu -1}\right) , \\ \frac{\partial {\underline{\psi }_{\times ,\mathrm {A}}}}{\partial {y}}(x,y,\varvec{\zeta },\varvec{\eta })&= \frac{1}{2}\left( \underline{\zeta }+\overline{\zeta } + (\mu +1)(\overline{\zeta }-\underline{\zeta }) \left( \tfrac{y-\underline{\eta }}{\overline{\eta }-\underline{\eta }} - \tfrac{\overline{\zeta }-x}{\overline{\zeta }-\underline{\zeta }}\right) \left| \tfrac{y-\underline{\eta }}{\overline{\eta }-\underline{\eta }} - \tfrac{\overline{\zeta }-x}{\overline{\zeta }-\underline{\zeta }}\right| ^{\mu -1}\right) . \end{aligned}$$