BUY THIS CABLE: http://www.amazon.com/Cable-Matters-SuperSpeed-2048x1152-1920x1200/dp/B00BPEV1XK
I've had this problem with my external monitor since I installed the 10.9.2 update. Tried PRAM resets, time-machined (fruitlessly) back to 10.9.1, swapped out kext files (computer gave me white screen of death), bought different mini-dvi cables, sent Apple bug reports, made threats, cried myself to sleep, etc. NOTHING WORKED-- short of buying a new imac because I'm a bajillionare with plenty of time and money to throw around.
After about 4+ months of not having an external monitor, pretty much everyone knew of my problem because it was always my wet blanket that I'd throw into a conversation when work came up. Had a friend recommend me this cable because it turns an available USB 3.0 port into an HDMI/DVI output. High resolution, video on it lags a little, colors might be a little wonky so basically: 1. Don't play video on it and expect a good time, 2. Don't print preview stuff on it.
I haven't tried the Yosemite update yet and have only heard wishful rumors that external monitors work now. For $50 however, this adapter is a cheap and usable fix to end the frustration of having a huge blank screen burning into the side of your face every time you're working on your computer.
Hope this info helps. Let me know if you have the same success!