  We present a hybrid optimization framework for a class of problems,
formalized as a generalization of the Continuous Energy-Con\-strained
Scheduling Problem (CECSP), introduced by Nattaf et al. (2014). This class is
obtained from challenges concerning demand response in energy networks. Our
framework extends a previously developed approach. A set of jobs has to be
processed on a continuous, shared resource. Consequently, a schedule for a job
does not only contain a start and completion time, but also a resource
consumption profile, where we have to respect lower and upper bounds on
resource consumption during processing. In this work, we develop a hybrid
approach for the case where the objective is a step-wise increasing function of
completion time, using local search, linear programming and O(n) lower bounds.
We exploit that the costs are known in the local search and use bounds to
assess feasibility more efficiently than by LP. We compare its performance to a
mixed-integer linear program. After that, we extend this to a hybrid
optimization framework for the General CECSP. This uses an event-based model,
and applies a decomposition in two parts: 1) determining the order of events
and 2) finding the event times, and hence the start and completion times of
jobs, together with the resource consumption profiles. We argue the broad
applicability of this framework.
