  We consider the multiple quantile hedging problem, which is a class of
partial hedging problems containing as special examples the quantile hedging
problem (F{\"o}llmer \& Leukert 1999) and the PnL matching problem (introduced
in Bouchard \& Vu 2012). In complete non-linear markets, we show that the
problem can be reformulated as a kind of Monge optimal transport problem. Using
this observation, we introduce a Kantorovitch version of the problem and prove
that the value of both problems coincide. In the linear case, we thus obtain
that the multiple quantile hedging problem can be seen as a semi-discrete
optimal transport problem, for which we further introduce the dual problem. We
then prove that there is no duality gap, allowing us to design a numerical
method based on SGA algorithms to compute the multiple quantile hedging price.
