We consider the problem of hedging the loss of a given portfolio of derivatives using a set of more liquid derivative instruments. We illustrate why the typical mathematical formulation for this hedging problem is ill-posed. We propose to determine a hedging portfolio by minimizing a proportional cost subject to an upper bound on the hedge risk; this bound is typically slightly larger than the optimal hedge risk achievable without cost consideration. We illustrate that the optimal hedging portfolio obtained by the proposed method is attractive since it consists of fewer instruments with a comparable risk. Finally we illustrate the importance of modeling volatility uncertainty in hedge risk minimization.