% file is s29.tex
The  {\em max-min eigenvalue problem} is:
\[
~~~(MMP)~~~\omega^*:=\max_{v^te=0} \lambda_{\min} (C - \Diag(v)).
\]
It can be formulated as:
\[
\omega^*=\max \{\omega : v^te=0,~C - \Diag(v)  \succeq  \omega I \}.
\]
With $y=v+\omega e,$ an equivalent problem is:
\[
\max \{e^ty: ~C - \Diag(y)  \succeq  0 \}.
\]
The dual  is the min-max of the Lagrangian
\[  \min_{X^a \succeq 0} \max_{y}
 ~e^ty+ \trace X(C-\Diag(y)). \]
or:
\[
\begin{array}{llll}
            \min & \trace CX& \\
{\rm subject~to}
              & \diag(X)= e \\
& ~X\succeq 0.
\end{array}
\]