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Yes.
You can find much of this out by trying
sage: g.traveling_salesman_problem??
to read the code. Note that in this case the function itself is in a .py (uncompiled, though in byte code I think) file. But the line
from sage.numerical.mip import MixedIntegerLinearProgram
indicates it uses MILP, and this is indeed a compiled file (as you can see by doing
sage: sage.numerical.mip.MixedIntegerLinearProgram?
where it indicates an .so file type.
However, in truth this calls an outside program (which you can set, e.g. CPLEX) to do the LP, so the compiled piece is less relevant than the fact that it uses a dedicated library to do the LP.
