If I don't show up for the first section of a round trip flight, will my second flight be cancelled?

I booked a round trip flight based on the assumption I would be leaving from one destination, but now I would prefer to leave from a different city. I could pay the airline to change my flight, but it would be significantly cheaper just to book a new one way flight and leave my existing itinerary as is (that is, missing the first flight on my original itinerary, but still using the same return flight).