New tenant breaking lease by smoking on premises - how to handle

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My online ad for a unit that was recently available to rent stipulated that there is no smoking of any kind allowed anywhere on the premises (including the porch stoop, back yard, and driveway).  When asked during the phone interview and again on the rental application whether he was a smoker, the tenant responded no. 

My bad for renting to him because he smelled like smoke at the showing, but I figured it was a one-off thing and that he wouldn't smoke on the premises.  Well of course he's a smoker and has been smoking on the stoop.  I know this because I live in the duplex unit next door and with my front door open, the smell is overpowering.  When I walk outside to see who is smoking, no one is anywhere, except the tenant is on his stoop with a ceramic mug next to him.  No cigarette in hand, because as soon as he sees my screen door open, he stubs out his cig.

When they (the smoker guy and his girlfriend) signed the lease, I reiterated to them very strongly that there is no smoking allowed anywhere on the premises, and they initialed that page of the lease that includes the no smoking clause, in italics.  

Whether or not you agree that it's reasonable not to allow tenants to smoke on the stoop is not the question.  The question is, he's only been living there a week and there are already lease violations (this is not the first time I've smelled smoke and seen him on the stoop--the last time, I asked him to walk down the street).  I am also concerned that they have an unauthorized tenant, as I've seen someone not on the lease enter with a key.  

Lesson for me is not to rent to someone who smells like smoke, but if they say on the application that they don't smoke, and they tell me they don't smoke, and they meet all my other requirements, it's a gray area for charges of discrimination.  

Anyway, now I could use advice about how to proceed.  I don't think I'll ever catch him in the act.  My concern if I tell him not to smoke on the stoop is that he'll just smoke inside, which is worse.  And come the winter, he'll probably smoke inside.  

Do I give a written warning?  A formal notice stating that he violated the lease?  I already did an inspection 2 days ago, having given 24 hours' notice, because I "needed to change out the shower arm" (which I did).  I saw a lighter on his bed and a bong in the other bedroom but no cigarette butts.  I should have checked the trash.  

The unit is a diligently restored and very nice midcentury-modern 3/1 in a duplex with their own fenced yard and garage for about $2k.  They have a 1 year lease and have put down a $2250 security deposit.  @Marcia Maynard or anyone else, any ideas?