Research
Research in
the topic of largescale unconstrained
nonlinear optimization where the function
had a special structure. The research
involved developing and analyzing
numerically stable algorithms for
minimizing a nonlinear function whose
Hessian (second derivative) matrix
is sparse (has many zero elements).
The implementation of these algorithms
required a thorough knowledge of software
design, data structure and numerical
analysis. Some of the situations in
which numerical analysis can help
include:
 Determination
of the numerical stability of a
computer generated solution. That
is, is the answer unstable because
of the inherent nature of the formulation
or because of the scaling of the
problem?
 Determination
of the numerical stability of the
algorithms.


TOX
RISK
TOX
RISK is a software package designed
to perform healthrelated risk assessments.
For example, it can be used to estimate
the likelihood of developing cancer
after exposure to different levels
of certain chemicals. The program
uses the maximum likelihood method
to estimate the parameters associated
with the risk. The underlying optimization
problem is a nonlinear program with
linear constraints. SBSI’s work
with TOX RISK involved examination
of the numerical properties of the
algorithm used by TOX RISK to perform
this optimization. The first step
in this task was to make sure that
the algorithm is stable. This step
involved examining the systems of
equations that must be solved during
the algorithm and verifying that the
procedures used to compute their solutions
limit the accumulation of roundoff
error. Otherwise, the computed solution
to the optimization problem may be
wrong. SBSI staff also checked that
the algorithm took precautions against
illconditioned problems, another
potential cause of incorrectly computed
solutions. In addition to verifying
the stability of TOX RISK’s
equation solver, SBSI staff also examined
the suitability of the algorithm used
to perform the optimization. Information
about stability, choice of numerical
tolerances and impact of numerical
errors are some of the issues that
must be dealt with properly to ensure
that an algorithm computes a true
optimal solution. 



