Primal-dual interior-point method

A primal-dual interior-point method solves perturbed KKT conditions while updating both the primal variables and Lagrange multipliers. For inequalities written as $c_i\ge 0$, a logarithmic-barrier path has positive $c_i,y_i$ and perturbed complementarity $c_i{y}_i=\mu$. A Newton system for these coupled equations gives primal and dual corrections together, with damping to preserve positivity. This can avoid explicitly forming the most ill-conditioned eliminated barrier Hessian, although scaling and regularity still matter.