The boost developers really loathe people who want to use DESTDIR, their ticket 1811 is still open :https://svn.boost.org/trac/boost/ticket/1811
Fortunately, the last addition there works:
./bootstrap.sh --prefix=$PREFIX ... ./b2 --prefix=$DESTDIR$PREFIX install
(obviously, expand $PREFIX to /usr and $DESTDIR to /some/path/to/directory)
If you wish to get verbose output from bjam, add '-d+2' to the invocation. Doing that, you can see that only g++ is used, and that the default compile flags are -O3. If you wish to append any other CXXFLAGS, add cxxflags="-fwhatever" to the first bjam invocation (the second invocation for install only copies the files).
For the little it is worth, the output of bootsrap.sh is in bootstrap.log - it uses gcc without any CFLAGS, but optimizing that seems a waste of time.