KKT Matrix for Equality-Constrained QPs

$$\left[\begin{array}{cc}G& -A^T\\ A& 0\end{array}\right]\left[\begin{array}{c}{x}^{\ast }\\ {\lambda }^{\ast }\end{array}\right]=\left[\begin{array}{c}-c\\ b\end{array}\right]$$

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Which can be rewritten by expressing $x^{\ast }$ as

$$x^{\ast }=x+p$$

Where $x$ is some stimate of the solution, and $p$ is the desired step.

We then get

$$\left[\begin{array}{cc}G& A^T\\ A& 0\end{array}\right]\left[\begin{array}{c}-p\\ {\lambda }^{\ast }\end{array}\right]=\left[\begin{array}{c}g\\ h\end{array}\right]$$

Where

$$h=Ax-b,g=c+Gx,p=x^{\ast }-x$$