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{\large \bf A Limited Memory Variable Metric Method in Subspaces and 
Bound Constrained Optimization Problems\footnotemark}

%{\large \bf A Limited Memory Variable Metric Method for Bound Constrained
% Optimization Problems\footnotemark}

{\bf Steven J. Benson  and
Jorge J. Mor\'e}

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% I THINK WE NEED TO EXPAND THE ABSTRACT AND FIRST PARAGRAPH TO
% STATE THAT WE HAVE RESULTS PERTAINING TO REDUCED SPACE BFGS METHODS
% IN GENERAL.  THEN WE USE THIS THEORY TO PROVIDE INSIGHT INTO OUR METHOD

% ALSO THE TITLE IS VAGUE.  IT IS ALMOST THE SAME TITLE USED BY
% BYRD, NOCEDAL, ET AL FOR THEIR TECHNICAL REPORT


\footnotetext
{ This work was supported by the Mathematical, Information, and
Computational Sciences Division subprogram of the Office of Advanced
Scientific Computing, U.S. Department of Energy, under Contract
W-31-109-Eng-38.}

\begin{abstract}

We describe an algorithm for solving nonlinear optimization problems
with lower and upper bounds that constrain the variables.  The algorithm
uses projected gradients to construct a limited memory BFGS matrix
and determine a step direction.  The algorithm has been implemented
and distributed as part of the Toolkit for Advanced Optimization (TAO).
We include numerical results demonstrate is effectiveness 
on a set of large test problems and
its scalability to multiple processors.

\end{abstract}

%  \noindent
%  {\bf Key words. } 
%  large-scale optimization,
%  gradient projection,
% limited memory method
% bound constrained optimization
% nonlinear optimization
% quasi Newton methods,
% BFGS methods,
% reduced space methods

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