Goal: Optimize convex function $f(\vec{x})$ subjected to constraint $A\vec{x} = \vec{b}$ starting at a point $\vec{x}_0$ that satisfies the constraint.
The problem only has equality constraint. Why does the solution requires using the KKT condition, which is for inequality constraint?
This lecture note mention using KKT condition and quadratic approximation gives the following:
Newton's method with line search:
$$ \begin{bmatrix} \nabla^{2} f & A^{T} \\ A & 0 \\ \end{bmatrix} \begin{bmatrix} u \\ w \\ \end{bmatrix} = \begin{bmatrix} - \nabla f \\ 0 \\ \end{bmatrix} $$
$\vec{x}^{(n+1)} = \vec{x}^{(n)} + t\vec{u}$ where $t$ is the step size found by backtracking.