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{\large Andrew R Conn \\
{\bf Bilevel Derivative-Free Optimization and its Application to Robust Optimization }}

IBM T J Watson Research Center \\ 1101 Kitchawan Rd Yorktown Heights NY 10598 USA
\\
{\tt arconn@us.ibm.com}\\
Luis N. Vicente\end{center}

We address bilevel programming problems when the derivatives of both the upper
and the lower level objective functions are unavailable.

The core algorithms used for both levels are trust-region interpolation-based
methods, using minimum Frobenius norm quadratic models when the number of
points is smaller
than the number of basis components. We take advantage of the problem structure to derive
conditions (as far as is possible closely related to the global convergence theory of the
underlying trust-region methods) under which the lower level can be solved inexactly and
sample points can be reused for model building.
In addition, we indicate numerically how effective these expedients can be.
A number of other issues are also discussed, from the extension to linearly constrained
problems to the use of surrogate models for the lower level response.

One important application of our work appears in the robust optimization
of simulation-based
functions. The robust counterpart of an optimization problem without derivatives falls
in the category of the bilevel problems under consideration here. We provide numerical
illustrations of the application of our algorithmic framework to such
robust optimization examples.


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