lower_bound on a vector of pair after checking one value of pair

I have a vector of pairs vector< pair< string,int> > in C++ and i want to lower_bound on string value but with additional constraint that the second value of pair should be less than or equal to a given value. Currently i am doing it using a compare template

bool compare(const T &a,const T &b){
if (a.first<=b.first&&b.second<=a.second) return true;
}

but it is not working properly. The vector is sorted according to the first value of pair. Example->The vector has following contents:

abcd,1
abcde,4
abcdex,3
abce,2

and i want to lower_bound on abc,3 so it should return abcd,1 but it is returning abcdex,3.Please help.