Recipe

Take one primal-dual interior-point step

Use when: you already have a strictly feasible point and want to advance the central path without fully re-solving a barrier subproblem from scratch

Goal: an updated (x, λ) pair, closer to the KKT solution, at a lower cost than Section 4's full resolve

Quick recipe

1
Combine stationarity $\nabla f(x)+\lambda\nabla g(x)=\mathbf0$ with perturbed complementary slackness $\lambda\, g(x)=-\mu$ .

Example

For this chapter's problem: $2x-\lambda=0$ , $2y-\lambda=0$ , and $\lambda(4-x-y)=-\mu$ .

Why: this is exactly the exact KKT system with one right-hand side nudged by μ instead of held at zero.
2
Treat $(x,\lambda)$ jointly as the unknown and take one Newton step (Chapter 4) on this system from the current point.

Why: one linear solve is far cheaper than fully re-minimizing a fresh barrier subproblem.
3
Move $(x,\lambda)$ in the Newton direction, shortened if necessary so $\lambda$ stays positive and $x$ stays strictly feasible.

Why: an undamped Newton step can overshoot past the boundary the same way it can in the purely unconstrained case (Chapter 4).
4
Reduce $\mu$ by a fixed factor and take another primal-dual step from the new point.
- the perturbed system's residual is small enough — stop; report $(x,\lambda)$ .
- it isn't — repeat from the linearization step.
Why: shrinking μ steadily walks the perturbed system toward the exact KKT system.

Common mistakes

Skipping the damping step and letting $\lambda$ go negative, which breaks dual feasibility mid-solve.
Shrinking μ before the current Newton step has actually reduced the residual, chasing a moving target.
Treating λ as an afterthought instead of an unknown solved for jointly with x — that joint solve is this method's entire advantage over Section 4.