Following up on the new criterion introduced by Abergel and Millot for hedging contingent claims in incomplete markets, we extend the approach to the case where there are transaction costs on the stock component. The local risk is a convex function of the local costs process. We derive corresponding optimal strategies in both discrete time and continuous time settings. Finally we give an application of our hedging method in the stochastic volatility case as well as in the jump diffusion case.