Thanks to some great SE answers on a previous question I posted, I realized that even though I don't have an RA at the moment (I am a first year PhD student), I must start my research ASAP. I am convinced that this is great advice and want to follow it.
But the concept of 'doing research' is still not perfectly clear to me. I am interested in convex optimization, that's what I intend to do my PhD in. So as of now, for my thesis, I have this vague idea in my mind that I'd solve a problem by maybe coming up with a cool algorithm and give some nice proofs using convex analysis.
But exactly how to start is a mystery. Here's what I think I should do: read a paper that looks interesting to me (this could be because of the abstract, or maybe it's a paper that has suddenly got very popular, or it could simply be based on the author's reputation, etc), thoroughly understand their results by working out proofs of all their statements, and then try to see what happens in slightly different cases than what they mention. Maybe relax some assumptions they make; or go the other direction, add a constraint and see if the algorithm can be improved because of this constraint, etc.
Is this how it's done? I feel so lost right now. When I try doing this, I have so far (in six months) not been able to really think of anything new. Even the existing proofs that I get are so complex, I can't think of any way to come up with something new. Can someone please give me some tips on how one gets started in theoretical research - how to come up with a problem, and how to proceed from there?
EDIT_ Adding more information: I don't have an adviser yet, and this is the post I am referring to in the first paragraph.