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Demand response (DR) is very important in the future smart grid, aiming to encourage consumers to reduce their demand during peak load hours. However, if binary decision variables are needed to specify start-up time of a particular appliance, the resulting mixed integer combinatorial problem is in general difficult to solve. In this paper, we study a versatile convex programming (CP) DR optimization framework for the automatic load management of various household appliances in a smart home. In particular, an L1 regularization technique is proposed to deal with schedule-based appliances (SAs), for which their on/off statuses are governed by binary decision variables. By relaxing these variables from integer to continuous values, the problem is reformulated as a new CP problem with an additional L1 regularization term in the objective. This allows us to transform the original mixed integer problem into a standard CP problem. Its major advantage is that the overall DR optimization problem remains to be convex and therefore the solution can be found efficiently. Moreover, a wide variety of appliances with different characteristics can be flexibly incorporated. Simulation result shows that the energy scheduling of SAs and other appliances can be determined simultaneously using the proposed CP formulation.