Core algorithm

Every QP qpax solves has the form

\[ \underset{x}{\text{minimize}} \quad \tfrac{1}{2}\,x^{\top} Q x + q^{\top} x \quad \text{s.t.} \quad A x = b, \;\; G x \leq h, \quad Q \succeq 0. \]

Introducing a slack \(s \geq 0\), equality multiplier \(y\), and inequality multiplier \(z \geq 0\), the KKT conditions are

\[ \begin{aligned} Q x + q + A^{\top} y + G^{\top} z &= 0, \\ G x + s - h &= 0, \\ A x - b &= 0, \\ s_i z_i &= 0, \qquad s, z \geq 0. \end{aligned} \]

The PDIP idea is to replace the hard complementarity \(s_i z_i = 0\) with a smooth condition \(s_i z_i = \mu\), then drive \(\mu \to 0\). Each iteration:

1. Form residuals. Evaluate the KKT residual at the current iterate.
2. Factor the KKT system. Reduce the saddle-point system to a smaller positive-definite system in \(x\) using the elimination from cvxopt, and factor it (Cholesky by default, QR on request).
3. Compute a Newton step. Solve the reduced KKT system for the step directions \((\Delta x, \Delta s, \Delta z, \Delta y)\).
4. Choose a step size. Pick \(\alpha \in (0, 1]\) so that the slack/dual variables stay strictly positive.
5. Update. Take a step of length \(\alpha\) along the Newton direction.

This skeleton is shared by both backends. They differ in how the slack/dual variables are updated (step 5) and, as a consequence, in how the relaxed KKT point is reached and differentiated.