Constraints on the directivity reduce the available bandwidth of antennas. We develop here a formulation that determines the smallest Q-factor for an arbitrary shaped embedded antenna for a given partial or total directivity. An embedded antenna is an antenna that is located within a device often as part of a circuit board, where the main volume of the device is dedicated to non-antenna functions. We show that semi-definite relaxation can be used to predict the lowest Q-factor in an embedded antenna for a partial or a total directivity in a given direction, and that the bound is tight. The paper ends with an example.