I know some topics are close to this one, but I didn't find any answers really giving me some clue.
I am currently developing an algorithm which is in a complexity (iteration complexity) of $\mathcal{O}(n * m)$ where $m \leq n$, so let's assume a worst case complexity of $\mathcal{O}(n^2)$, even if it will probably never happen.
I want to compare the computational complexity of my algorithm with the algorithms used in optimization solvers such as CPLEX or Gurobi. I have been able to test directly my algorithm against CPLEX and Gurobi, and I am way faster. But still, I would like to have something more formal.
My question is the following: what are the best possible computational complexity for algorithms such as interior points methods and Second Order Cone Programming ?
For the first one, I found that the complexity is at best (State of the art method which is used in CPLEX and Gurobi), looking like $\mathcal{O}(n^3L)$ (computational complexity). I didnt find anything for the SOCP, so if you have any intels, please share.
Another question here is: when we talk about "complexity" in its globality, is it right to talk only of the highest value between the iteration complexity and the computational complexity (just what I have done) ? If not, how can we define a "global" value for both complexity ?
Thank you very much for your help.