Primal-dual methods find by Newton's method a point on the central path for a sufficiently large value of t. This way solutions are positive. Then t is decreased (and the system is solved again) in a way such that the trajectory of the solutions follows closely the central path. Note F (x, y, r) and Ft (x, y, r) have the same Jacobian, so one has to solve at each iteration step: Non singularity of the Jacobian: it is enough that rows of A are linearly independent. More critical the non singularity of the Jacobian in the optima (otherwise the method is not superlinear): unique and non degenerate optima guarantee non singularity. Non unique optima have a singular Jacobian, however the method does not converge to vertices but to strictly convex combinations of optimal vertices. Global convergence is assured by careful choice of t and the initial solution.

---