% file is s31.tex
PRIMAL-DUAL INTERIOR POINT  METHOD\\
The primal-dual log-barrier problem is then
\[
~~~(B)~~~~\begin{array}{llll}
     \omega^*_\mu= \max & e^ty + \mu \log \det (Z) &\\
{\rm subject~to}
              & C-\Diag(y) -Z = 0.
            \end{array}
\]
The Lagrangian $L(y,Z,X)$ is 
\[  e^ty +  \mu \log \det (Z) + \trace X(C-\Diag(y) -Z),\]
while the optimality conditions $F(y,Z,X)=0$ are:
\[
\begin{array}{rcl}
   e- \diag(X) &=& 0~~\mbox{(primal
                     feasibility)}\\
   -X + \mu Z^{-1} &=& 0~~\mbox{(complementary slackness)}\\
  C-\Diag(y) -Z &=& 0~~\mbox{(dual feasibility)}\\
  Z \succeq 0,~ X \succeq 0.
\end{array}
\]