How can you wipe and reinstall OSX on a Mac Mini (which has no optical drive)?
Without an optical drive, how can you clean the disks and still be
  able to reinstall an OS?


Simplest and cheapest way: create a bootable volume from the OS X Installer on a USB Stick.

2 days ago I used the (free) DiskMakerX to clone the 10.10.3 OS X Installer (it works also with older OS X versions) from AppStore to a USB Stick and used that bootable clone to start from it and install OS X on another volume. Everything worked fine.