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A Special Complementarity Function Revisited

Roger Behling (roger.behling***at***ufsc.br)
Andreas Fischer (Andreas.Fischer***at***tu-dresden.de)
Klaus Schönefeld (Klaus.Schoenefeld***at***tu-dresden.de)
Nico Strasdat (Nico.Strasdat***at***tu-dresden.de)

Abstract: 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.

Keywords: Karush-Kuhn-Tucker system, nonunique multipliers, degenerate solution, constrained Levenberg-Marquardt method

Category 1: Complementarity and Variational Inequalities

Category 2: Nonlinear Optimization

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

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Entry Submitted: 04/27/2018
Entry Accepted: 05/01/2018
Entry Last Modified: 06/08/2018

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