  In this paper, we aim to perform sensitivity analysis of set-valued models
and, in particular, to quantify the impact of uncertain inputs on feasible
sets, which are key elements in solving a robust optimization problem under
constraints. While most sensitivity analysis methods deal with scalar outputs,
this paper introduces a novel approach for performing sensitivity analysis with
set-valued outputs. Our innovative methodology is designed for excursion sets,
but is versatile enough to be applied to set-valued simulators, including those
found in viability fields, or when working with maps like pollutant
concentration maps or flood zone maps.
  We propose to use the Hilbert-Schmidt Independence Criterion (HSIC) with a
kernel designed for set-valued outputs. After proposing a probabilistic
framework for random sets, a first contribution is the proof that this kernel
is characteristic, an essential property in a kernel-based sensitivity analysis
context. To measure the contribution of each input, we then propose to use
HSIC-ANOVA indices. With these indices, we can identify which inputs should be
neglected (screening) and we can rank the others according to their influence
(ranking). The estimation of these indices is also adapted to the set-valued
outputs. Finally, we test the proposed method on three test cases of excursion
sets.
