I know that under Windows, there are API functions like global_alloc() and such, which allocate memory, and return a handle, then this handle can be locked and a pointer returned, then unlocked again. When unlocked, the system can move this piece of memory around when it runs low on space, optimising memory usage.
My question is that is there something similar under Linux, and if not, how does Linux optimize its memory usage?
Those Windows functions come from a time when all programs were running in the same address space in real mode. Linux, and modern versions of Windows, run programs in separate address spaces, so they can move them about in RAM by remapping what physical address a particular virtual address resolves to in the page tables. No need to burden the programmer with such low level details.
Even on Windows, it's no longer necessary to use such functions except when interacting with a small number of old APIs. I believe Raymond Chen's blog and book have some discussions of the topic if you are interested in more detail. Eg here's part 4 of a series on the history of GlobalLock.