The improved climb procedure is a technique to increase the normal V2 speed towards the optimum gradient speed.
The resulting gradient improvement can then be traded for higher weight , or sometimes better obstacle clearance.
With an increase in V2 , Vr has to be increased as well.
(Obviously it will require additional runway)
It sounds strange however to have a Vr higher then V2.
Allthough there seems to be
no requirement that V2
must be higher then Vr !
What airplane are you talking about anyway ?
Cheers