Alot of people including me have been having the same exact problem where their Alienware laptop wont turn on after you shut it down for no reason at all. Now, people DO NOT panic! There is nothing wrong with your hardware. The laptop is perfectly fine. What is causing this issue is the BIOS they released (1.2.12, 1.2.13, 1.2.14) these BIOS cause this problem ive never had an issues before on the latest 2015 bios that was installed on the Alienware when I purchased it until I updated the bios mentioned above. Alienware I please ask you to not release a bios that would cause issues to people and please do intensive tests before you release anything. The sticky they have on the forum here did fix the issue though, but it may come back.

this will solve your problem, in this link it says to install BIOS 1.2.13 or higher, BUT DO NOT DO THAT, and instead when the computer turns on after following the directions for the fix mentioned in the link below, download BIOS 1.2.8 and install it, then after having 1.2.8 you wont have the problem anymore.

http://www.dell.com/Support/Article/us/en/19/SLN301252