Have the same problem, push the start button, runs from 2 to 60 seconds, shuts down. Tried all the fixes suggested on this site, no joy. Took to the repair shop (twice) and they could not duplicate the problem after several days each time, so brought it home and worked o.k. for a couple of weeks and then started doing the same thing. The third time I took it to the shop,we plugged it in while I was there and it would not start. They ordered a power supply and I should have it back tomorrow. Glad I have applecare.
I sleep all three of my macs all the time.