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\title{A Case Study in the Performance and Scalability of
Optimization Algorithms\footnotemark}

\author{Steven J. Benson
\and
Lois Curfman McInnes
\and
Jorge J. Mor\'e \\
Mathematics and Computer Science Division \\
Argonne National Laboratory
}

\begin{abstract}
We analyze the performance and scalabilty of algorithms for the
solution of large optimization problems on 
high-performance parallel architectures.
Our case study uses the GPCG (gradient projection, conjugate gradient) algorithm
for solving bound-constrained convex quadratic problems.
Our implementation of the GPCG algorithm within the Toolkit for 
Advanced Optimization (TAO) is available
for a wide range of high-performance architectures
and has been tested on problems with over 2.5 million variables.
We analyze the performance as a function of the number of variables, 
the number of
free variables, and the preconditioner.
In addition, we discuss how the software
design facilitates algorithmic comparisons.
\end{abstract}

%  \noindent
%  {\bf Key words.} 
%  large-scale optimization,
%  high-performance architectures,
%  conjugate gradients, 
%  gradient projection,
%  quadratic programming., 

\category{G.4.m} {Mathematical Software} {Parallel and vector implementations}
\category{D.2.13} {Reusable Software} {Reusable libraries}
\category{G.1.6}{Optimization}{Quadratic Programming Methods}
\terms{Bound constrained optimization, Parallel Efficiency}
\keywords{}
\begin{bottomstuff}
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.
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\end{bottomstuff}
\markboth{S.J. Benson, L. McInnes, J.J. Mor\'e}
     {A Case Study in the Performance and Scalability of
Optimization Algorithms}
\maketitle


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