Shortfall risk minimisation vs. symmetric (quadratic) hedging

We deal with the seller of a contingent claim who wants to hedge against the corresponding risk by means of a self-financing strategy, endowing less initial capital than the one required for (super)hedging. Two classical criteria used in this situation are the shortfall risk minimisation and the symmetric hedging (a natural generalisation of the quadratic hedging problem). We show that these two problems are equivalent if the market is complete. The case when the market is incomplete is also discussed