\documentstyle[ltexconf]{report} 
\begin{document}
 
\title{Newton Directions and Least Squares for Semidefinite
Programming}
 
\abstract{Primal-dual 
interior-point methods have proven to be the solution method
of choice for semidefinite programming (SDP)
problems. Many of the techniques that have
been so successful for linear programming (LP)  have been extended to
SDP.
We discuss the similarities and differences between LP and SDP.
In particular,
one major difference between LP and SDP that raises
several interesting questions is the form of the
complementarity condition used in the optimality conditions,
i.e. $ZX - \mu I = 0.$
The primal matrix $X$ and the dual slack matrix $Z$ are diagonal
in LP but may only be
symmetric in SDP; this results in $ZX$ not being symmetric in
general. In addition, problems with numerical instability can
arise near the optimal solution.\\
There have been various approaches which attempt to symmetrize
the complementarity condition so that the linearization of the
optimality
conditions are ``square'', i.e. map between the same spaces. We consider
different approaches which require no symmetrization.
We concentrate on solving the ``nonsquare,'' or overdetermined, system
using nonlinear least squares. In particular, we use
Gauss-Newton type methods.
}


\author{\underbar{Henry Wolkowicz}\\
University of Waterloo, Waterloo, Ont. Canada\\
\\
}

\end{document}

% end of example file.

%Henry Wolkowicz                |Fax:   (519) 725-5441
%University of Waterloo         |Tel:   (519) 888-4567, 1+ext. 5589
%Dept of Comb and Opt           |email:  henry@orion.math.uwaterloo.ca
%Waterloo, Ont. CANADA N2L 3G1  |URL: http://orion.math.uwaterloo.ca/~hwolkowi


