This paper deals with optimal scheduling of demand response in a residential setup when the electricity prices are known ahead of time. Each end-user has a "must-run" load, and two types of adjustable loads. The first type must consume a specified total amount of energy over the scheduling horizon, but its consumption can be adjusted across the horizon. The second type of load has adjustable power consumption without a total energy requirement, but operation of the load at reduced power results in dissatisfaction of the end-user. Each adjustable load is interruptible in the sense that the load can be either operated (resulting in nonzero power consumption), or not operated (resulting in zero power consumption). Examples of such adjustable interruptible loads are charging a plugin hybrid electric vehicle or operating a pool pump. The problem amounts to minimizing the cost of electricity plus user dissatisfaction, subject to individual load consumption constraints. The problem is nonconvex, but surprisingly it is shown to have zero duality gap if a continuous-time horizon is considered. This opens up the possibility of using Lagrangian dual algorithms without loss of optimality in order to come up with efficient demand response scheduling schemes.