Superrelaxation and the rate of convergence in minimizing quadratic functions subject to bound constraints ∗ Zdenˇ ekDost´al,MartaDomor´adov´a,andMarieSadowsk´a † Abstract The paper resolves the problem concerning the rate of convergence of the working set based MPRGP (modified proportioning with reduced gradient projection) algo- rithm with a long steplength of the reduced projected gradient step. The main results of this paper are the formula for the R-linear rate of convergence of MPRGP in terms of the spectral condition number of the Hessian matrix and the proof of the finite termination property for the problems whose solution does not satisfy the strict com- plementarity condition. The bound on the R-linear rate of convergence of the projected gradient is also included. For shorter steplengths these results were proved earlier by Dost´al and Sch¨ oberl. The efficiency of the longer steplength is illustrated by numer- ical experiments. The result is an important ingredient in development of scalable algorithms for numerical solution of elliptic variational inequalities and substantiates the choice of parameters that turned out to be effective in numerical experiments. AMS classification: Primary 65K05; Secondary 90C20 Key words and phrases: quadratic programming, bound constraints, inexact active set strategy, rate of convergence, finite termination. Short title: Superrelaxation in minimizing quadratic functions * This research has been supported by the grants GA CR 201/07/0294, AS CR 1ET400300415, and the Ministry of Education of the Czech Republic No. MSM6198910027. † FEI V ˇ SB-Technical University Ostrava, Tˇ r 17 listopadu, CZ-70833 Ostrava, Czech Republic, E-mail: zdenek.dostal@vsb.cz, Phone: +420 596 995 227, Fax +420 596 919 597 1