%   file is s9.tex
{\bf THEOREM (DUALITY)}
\begin{itemize}
\item
$y^*$, $\lambda^*$, $\mu^* = \mu(y^*)$ optimal
\item
Lagrangian function
\[
L(y,\nu,\omega) = \mu(y) + \nu (\alpha - y^tCy) + \omega( y^tCy - \beta)
\]
\item
Lagrange dual functional
\[
\phi ( \nu,\omega) = \inf L(y,\nu,\omega)
\]
\item
quadratic dual functional 
\[
h(\nu,\omega) = \nu \alpha - \omega \beta - \psi^t (B-\nu C + \omega
C)^{-1} \psi
\]
\end{itemize}
Then
\[
\begin{array}{rcl}
\mu^*  &= & \max_{\nu \leq 0,\omega \leq 0} \phi (\nu,\omega)\\
& = & \sup_{\stackrel{B - \nu C + \omega C > 0}{\nu \leq 0,\omega \leq
0}}
h(\nu,\omega)
\end{array}
\]
%where the latter equality requires that the regular case holds.
%Moreover, the first equality is attained by
%\[
%  \nu^* = -(-\lambda^* )_+ {\mbox ~and~}
%          \omega^* = -(\lambda^* )_+
%\]