I'm currently looking into optimization, specifcally into primal-dual interior point methods to solve nonlinear, convex, and constrained optimization problems.
I face problems of type
$$\mathrm{min} f(\mathbf{x}) \\ \mathrm{s.t.} \; \mathbf{g}(\mathbf{x}) \leq \mathbf{0} \\ \mathbf{x} >= \mathbf{0} $$
usually they get converted to equaltiy contraints using slack varibales $\mathbf{s}$ as $$\mathrm{min} f(\mathbf{x}) \\ \mathrm{s.t.} \; \mathbf{g}(\mathbf{x}) + \mathbf{s} = \mathbf{0} \\ \mathbf{x} >= \mathbf{0} \;\; \mathbf{s} >= \mathbf{0} $$
Now I convert this problem into a barrier problem, e.g. $$\mathrm{min} f(\mathbf{x}) - \mu \sum_i \mathrm{log}(x_i) - \mu \sum_i \mathrm{log}(x_i) \\ \mathrm{s.t} \; \mathbf{g}(\mathbf{x} + \mathbf{s}) = 0$$
So far so good. When I set up the Lagrangian of this barrier problem i get
$$ \mathfrak{L}(\mathbf{x},\mathbf{\lambda},\mathbf{s}, \mu) = f(\mathbf{x}) - \mu \sum_i \mathrm{log}(x_i) - \mu \sum_i \mathrm{log}(x_i) - \mathbf{\lambda}^T (\mathbf{g}(\mathbf{x}) - \mathbf{s})) $$ From this perspective I understand that $\mathbf{x}$ is the vecotr of primal variables, $\mathbf{\lambda}$ is the vector of Langrange parameters and $\mathbf{s}$ is the vector of slacks.
To solve this system I apply a Newton scheme to the barrier system, yielding my iterates $(\mathbf{x}_k, \mathbf{\lambda}_k, \mathbf{s}_k$) for a fixed barrier parameter $\mu$ until the KKT conditions for the barrier system are fulfilled.
At this point I lack some understanding. First of all, I am not sure what my dual probelm is in this case. Since I converted the inequality constraints to equality constraints I a missing the respective vector of Langrange parameters (usually denoted as $\mathbf{\mu}, \mathbf{z}$ or $\mathbf{\nu}$). Instead I got an additional vector of slacks $\mathbf{s}$. Is $\mathbf{s}$ now part of the dual or not? As I understand it, it is separate. How is it related to the Lagrange parameters of the inequality constraints which are not present when I use slacks?
Second, I tried to calculate the next barrier parameter $\mu_{k+1}$ based on the complementary gap. Using the book "Numerical Optimization" by "Nocedal and Wright", they suggest using the complementary value calculated as $$ \mathbf{s}_k^T \mathbf{\lambda}_k $$ I do not understand why I need to use the slacks in this case. Intuitively it makes sense, since the slacks replace the inequality constraints but I don't understand exactly why. If someone could provide me a bit of context, it would be really useful to me.