Applications for optimization with uncertain data in practice often feature a possibility to reduce the uncertainty at a given query cost, e.g., by conducting measurements, surveys, or paying a third party in advance to limit the deviations. We model such situations by a class of decision-dependent robust optimization problems in which the uncertainty set can be modified elementwise at a cost. In our framework, uncertain cost coefficients lie in bounded intervals and the optimizer chooses a query vector that shrinks each interval towards a hedging point, possibly down to a single value. We refer to this overall modeling paradigm with decision-dependent uncertainty sets as optimization under elementwise controllable uncertainty (OCU). We study two different problem settings – one with known and one with unknown hedging points – in more detail, in which we handle the remaining uncertainty by the paradigm of robust optimization. For both settings, we draw connections to the existing literature, provide bounds on the optimal objective value, and give a single-level non-linear reformulation. Furthermore, we state assumptions under which the three- respectively four-level problem can be solved as a single-level mixed-integer linear program. Finally, we formalize the phenomenon of budget deflection, where a parameter is queried solely to control the uncertainty for other parameters. We provide examples illustrating when budget deflection can occur and identify modeling choices under which it is provably excluded.
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