ATH won't run

Hello,
I'm trying to run an Hardware Test on my Macbook Pro 15'' (mid 2010), but even following the instruction the AHT won't run.

I've been trying to:

• Press D and then turn on the MacBook, waiting till the gray screen comes up and then release;
• Press D and then turn on the MacBook, waiting till the apple shows up and then release;
• Press D and then turn on the MacBook, waiting till the loading in the apple screen was finished;
• Turn on the MacBook and press D, till the gray screen shows up;
• Turn on the MacBook and press D, till the apple shows up;
• Turn on the Macbook and press D, till the apple screen finishes the loading.

Nothing of the above has been working, and I'm wondering if I'm doing something strange.

I'm trying this because sometimes my Macbook will randomly restart (actually I noticed that it happens after using some gestures such as switching between screens, using mission control, pressing the space bar to enlarge a file preview or even putting in full screen mode a video on youtube or similar) and after checking with the Support by phone they told me that it may be because some hardware issues (since I formatted my machine yesterday and initialized, then installed the software by dvd and manually brought back just files such as images, docs, photoshop projects and so on but any system file).

So before to bring it to an Apple Store, I wanted to make sure that it's about an hardware issue and - if so - where is the issue.

Thank you!

In case it's needed:

MacBook Pro 15'' - Mid 2010 | OS X Yosemite (10.103)

Processor 2,53 Ghz Intel Core i5

Memory 4GB 1067 MHz DDR3

Graphics NVIDIA GeForce GT 330M 256 MB