Uh, ask them? If they refuse, just threatening them with a court case will probably do the job. They are legally obligated to uphold their end of your lease agreement.

That is not correct.

   Let L = the landlord,
       R = the tenant who is renting out his apartment,
       T = the tenant who is being annoyed by the
           problems R's sub-renters are causing

You are proposing that T sue L for not enforcing a term in R's lease against R. R does not wish to have this term enforced.

The lease agreement being violated (at least as far as the term that prohibits sub-renting the apartment goes) is the one between L and R, not the one between L and T. T would not have standing to sue since T is not a party to the contract between L and R.