  We introduce the notion of log R-maps, and develop a proper moduli stack of
stable log R-maps in the case of a hybrid gauged linear sigma model. Two
virtual cycles (canonical and reduced) are constructed for these moduli stacks.
The main results are two comparison theorems relating the reduced virtual cycle
to the cosection localized virtual cycle, as well as the reduced virtual cycle
to the canonical virtual cycle. This sets the foundation of a new technique for
computing higher genus Gromov-Witten invariants of complete intersections.
