We consider the problem of optimizing an unknown function given as an
oracle over a mixed-integer set. We assume that oracle is expensive to
evaluate, so that estimating partial derivatives by finite differences is
unpractical. In the literature, this is typically called a black-box
optimization problem with costly evaluation.
This problem class finds many application. Our main motivation comes from
a problem of architectural design. More precisely, lighting and heating
conditions of a building depend on the value of some design parameter
(these are the decision variables in the usual optimization paradigm).
The designer uses simulation software that takes as input the value of the
design parameters and outputs illuminance and temperature values inside
the building. In many cases, running a simulation is very time consuming.
Since we do not know explicitly the relationship between the design
parameters and the simulation output, we cannot define an objective
function in terms of the decision variables.
To solve this problem, we use the Radial Basis Function (RBF) interpolation
approach. Given some evaluations provided by the oracle, an approximation
of the objective function using the RBFs can be computed, leading to the
so called "surrogate" model. The quality of the solution found by solving
the resulting problem depends on the quality of the surrogate model.
In general, one would like to find a good surrogate model by minimizing
the number of expensive evaluations of the oracle.

We present the main features of the RBF method and some research questions
still open, as well as some details about the architectural design
application we are studying.