Ubuntu – Upgrade popup says the graphic hardware is not supported. Is this true

I think you would have to download and install AMDs proprietary driver from their website.

For 32 and 64-bit: http://support.amd.com/us/gpudownload/linux/Pages/radeon_linux.aspx?type=2.4.1&product=2.4.1.3.42&lang=English

after downloading right click on the file and mark it as executable under properties. Now open a terminal and type in

sudo sh /home/your_username/Downloads/amd-driver-installer-12-2-x86.x86_64.run

the installer will guide you through the installation.
After the installation was succesful, type into the terminal

sudo aticonfig --initial -f

and finally

sudo reboot

I'd like to share my experience with a similar issue.

I recently bought a laptop with the same processor and I was experiencing problems with graphics. I necessarily have to work with Ubuntu 14.04 but it was not recognizing my laptop's hardware (skylake not supported). I had to install manually kernel 4.4.6 so everything worked properly, however I noticed compiz was taking more than 60% cpu. glxgears showed errors like: "i965_dri.so does not support the 0x1916 PCI ID" "libGL error: pci id for fd 4: 8086:1916, driver (null)", etc.. and framerates above 300.

The reason was the GPU drivers were not properly updated so the CPU has to do the job in its place. To solve it I ran the following commands:

sudo apt-get install --install-recommends linux-generic-lts-wily xserver-xorg-core-lts-wily xserver-xorg-lts-wily xserver-xorg-video-all-lts-wily xserver-xorg-input-all-lts-wily libwayland-egl1-mesa-lts-wily

which are installed along with kernel 4.2.

After reboot with my working kernel 4.4.6 glxgears shows proper framerates (around 60 FPS) and compiz CPU consumption is normal now.

I'm a newbie here but I'm happy to have managed to solve this problem, I hope this is also useful for others.