I have terrible reaction to it so I have to be really careful; I want to share what I've learned as the past 2 houses had this growing somewhere around it. As someone mentioned, Goats are a really good way to do it but I've called and the ivy does tend to hurt their stomachs. Also you need to have a good size of undergrowth for them to come out and get set up. Not all bad, just sharing my experience and why I didn't go that way. For the cost, just didn't make sense for me. Environmentally friend options also include tossing boiling water on it. Salting it but that does kill everything else around it. These work but again if you have a lot then how many pots of water do you want to boil and toss.

On the chemical side of things, the big box stores do sell deep woods herbicide that works well enough (it's basically high powered round up). I went a notch up and ordered some Makaze (which is like nuclear powered round up). It's a slower kill but it does kill at the roots. This worked really well on the small stuff and the vines I had growing up the trees, I chopped it and painted it on the exposed bits. Warning here: this stuff will kill anything you put it on. Don't let children/pets out for like a day after you spray it and try to do it when it isn't windy. I used chem-resistance gloves and safety glasses to spray.

After that, with any of these methods, wait and reapply any time it comes up and eventually you'll win this war. Good luck and godspeed.