When solving a reduced KKT system of a nonlinear (and nonconvex) constrained program after eliminating slack and dual variables, how do we actually take the next step in a primal-dual method?
For example, following notation from NW, if the original nonlinear system is like (19.12) \begin{align} \begin{bmatrix} \nabla_{xx}^2 L & 0 & A_E(x)^T & A_I(x)^T \\ 0 & \Sigma & 0 & -I \\ A_E(x) & 0 & 0 & 0 \\ A_I(x) & -I & 0 & 0 \\ \end{bmatrix} \begin{bmatrix} p_x \\ p_s \\ -p_y \\ -p_z \end{bmatrix} = -\begin{bmatrix} \nabla f(x) - A_E(x)^Ty - A_I(x)^T z \\ z - \mu S^{-1}e \\ c_E(x) \\ c_I(x) - s \end{bmatrix}, \end{align}
then I see how a solution gives us a way to update $x,s,y,z$. However, if we solve a reduced system of the form
\begin{align} \begin{bmatrix} \nabla_{xx}^2 L + A_I(x)^T\Sigma A_I(x) & A_E(x)^T \\ A_E(x) & 0 \end{bmatrix} \begin{bmatrix} p_x \\ -p_y \end{bmatrix} = \text{?} \end{align}
then (1) what is the RHS; and (2) how do we update $s,z$?
EDIT: Can I have some help on the details for how we eliminate variables in moving from the larger system to the smaller system?