We call an optimization problem an \emph{optimization problem with controllable uncertainty} if a) it contains uncertain input data and b) prior to deciding the optimization variables, the optimizer can for a certain cost reduce the scenario set of this uncertain data.

In particular, we are interested in situations where each uncertain parameter is a priori known to lie in a closed, bounded interval and the optimizer has a fractional choice to shrink each interval.

We study this setting for optimization problems with uncertain non-negative linear cost functions and non-negative decision variables.

Moreover, we assume the scenario set to be restricted by some polyhedron, e.g., we consider so-called budgeted uncertainty. In particular, we show that the resulting three-level problem can be solved as a single mixed integer program for binary queries, if the underlying optimization problem is a linear program.

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