Apparatus, systems, methods, and related computer program products for optimizing a schedule of setpoint temperatures used in the control of an HVAC system. The systems disclosed include an energy management system in operation with an intelligent, network-connected thermostat located at a structure. The thermostat includes a schedule of setpoint temperatures that is used to control an HVAC system associated with a structure in which the thermostat is located. The schedule of setpoint temperatures is continually adjusted by small, unnoticeable amounts so that the schedule migrates from the original schedule to an optimal schedule. The optimal schedule may be optimal in terms of energy consumption or some other terms.