Differentiable Local Barrier-penalty Paths Barrier-penalty Paths

Perturbations of Karush-Kuhn-Tucker conditions play an important role for primal-dual interior point methods. Beside the usual logarithmic barrier various further techniques of sequential unconstrained minimization are well known. However other than logarithmic embeddings are rarely studied in connection with Newton path-following methods. A key property that allows to extend the class of methods is the existence of a locally Lipschitz continuous path leading to a primaldual solution of the KKT-system. In this paper a rather general class of barrier/penalty functions is studied. In particular, under LICQ regularity and strict complementarity assumptions the differentiability of the path generated by any choice of barrier/penalty functions from this class is shown. This way equality as well as inequality constraints can be treated directly without additional transformations. Further, it will be sketched how local convergence of the related Newton path-following methods can be proved without direct applications of self-concordance properties. keywords: Perturbed KKT-systems, general barrier-penalty embedding, differentiable path, path-following methods, interior point methods 1 Barrier/penalty functions and primal-dual paths Barrier/penalty methods and its path-following variants form an important class of numerical methods for constrained optimization problems via a family of unconstrained ones (cf. [2], [8], [9]). While large classes of classical ∗[email protected] TU Dresden, Institut für Numerische Mathematik, D01062 Dresden, Germany; Paper written with financial support of DFG grant GR 1777/2-2