\begin{equation}
\label{equality-duality}
\gamma_M
~\stackrel{\mathrm{(A)}}{=}~
\sup_{\{\mathfrak{m}^+_{k,m}\}} \inf_{\substack{\{\lambda_m\} \\ \{ \mu^+_{k,m} \} }} \mathfrak{L}
~\stackrel{\mathrm{(B)}}{=}~
\inf_{\substack{\{\lambda_m\} \\ \{ \mu^+_{k,m} \} }} \sup_{ \{\mathfrak{m}^+_{k,m}\} } \mathfrak{L}
~\stackrel{\mathrm{(C)}}{=}~
\gamma^\ast_M,
\end{equation}
