Recently, a local framework of Newton-type methods for constrained systems of equations has been developed which, applied to the solution of Karush-KuhnTucker (KKT) systems, enables local quadratic convergence under conditions that allow nonisolated and degenerate KKT points. This result is based on a reformulation of the KKT conditions as a constrained piecewise smooth system of equations. It is an open question whether a comparable result can be achieved for other (not piecewise smooth) reformulations. It will be shown that this is possible if the KKT system is reformulated by means of the Fischer-Burmeister complementarity function under conditions that allow degenerate KKT points and nonisolated Lagrange multipliers. To obtain this result, novel constrained Levenberg-Marquardt subproblems are introduced which allow significantly longer steps for updating the multipliers. Based on this, a convergence rate of at least 1.5 is shown.
Citation
Optimzation, https://doi.org/10.1080/02331934.2018.1470177, Technical Report: MATH-NM-06-2017, Institute of Numerical Mathematics, Faculty of Mathematics, TU Dresden, Germany, December 2017