So I noticed that equality constraints cannot be treated with interior barriers, which means we can only work with inequality constraints. I know that a trick sometimes used is to double the amount of inequality constraints by keeping $g_i(x) \le 0$ and generating new $-g_i^{new} \ge 0$. However I am guessing this will not work with interior barrier algorithms right?
This makes me wonder whether this also applies to the "basic interior point method?" for which i have included a photo of the algorithm (not sure if there is a more specific name for this provided algorithm)?
So to be more specific with my question, can i also use this algorithm using only equality constraints? If i can, does this mean we remove the 2nd column and 2nd row from the matrix in step b), as well as the 2nd entry in the left and right hand vector? Or do we simply set all inequalities $g_i(x) \le 0$ as being nothing (zero) and move along as if nothing were different?
I am trying to get this to work with my optimization problem $$ \min_{\theta \in \mathbb{C}^{n} } Tr((\textbf{HH}^H)^{-1}) \; \; s.t. \; \; Re^2(\theta_i)+Im^2(\theta_i) -1 =0 \; \; \forall i \in \{1,\ldots,64\} $$ Where $\textbf{H}$ is a linear function of $\theta$, and $n=64$
Super appreciate any help, I am a total noob to this stuff.
Equality constraints are naturally supported by IPM. They simply appear "as is" in the KKT conditions and linearized in the linear system (augmented or KKT system) resulting from the Newton step.
This is actually true for all Newton-based methods (IPM, SQP, etc). The difference is how they handle inequality constraints. Modern IPM add slacks (which results in equality constraints only), then use a barrier term for the bounds (original variables + slacks). In modern formulations, all the iterates are interior wrt the primal bounds (original variables + slacks) and the dual bounds, not wrt the original constraints.
Edit: I'm guessing that $h$ in your screenshot are equality constraints.