Abstract
A new active set based algorithm is proposed that uses the conjugate gradient method to explorethe face of the feasible region defined by the current iterate and the reduced gradient projection with the fixedsteplength to expand the active set. The precision of approximate solutions of the auxiliary unconstrained problemsis controlled by the norm of violation of the Karush-Kuhn-Tucker conditions at active constraints and the scalarproduct of the reduced gradient with the reduced gradient projection. The modifications were exploited to find therate of convergence in terms of the spectral condition number of the Hessian matrix, to prove its finite terminationproperty even for problems whose solution does not satisfy the strict complementarity condition, and to avoidany backtracking at the cost of evaluation of an upper bound for the spectral radius of the Hessian matrix. Theperformance of the algorithm is illustrated on solution of the inner obstacle problems. The result is an importantingredient in development of scalable algorithms for numerical solution of elliptic variational inequalities.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
O. Axelsson, Iterative Solution Methods. Cambridge University Press: Cambridge, 1994.
M.S. Bazaraa and C.M. Shetty, Nonlinear Programming. J. Wiley: New York, 1979.
R.H. Bielschowski, A. Friedlander, F.A.M. Gomes, J.M. Martýnez, and M. Raydan, "An adaptive algorithm for bound constrained quadratic minimization," Investigacion Operativa, vol. 7, pp. 67–102, 1997.
P.H. Calamai and J.J. Moré, "Projected gradient methods for linearly constrained problems," Math. Program-ming, vol.39 pp. 93–116, 1987.
T.F. Coleman and L.A. Hulbert, "A globally and superlinearly convergent algorithm for convex quadratic programs with simple bounds," SIAM J. Optim, vol. 3, no. 2, pp. 298–321, 1993.
R.S. Dembo and U. Tulowitzski, "On minimization of a quadratic function subject to box constraints," Working Paper No. 71, Series B, School of organization and Management, Yale University: New Haven, CT, 1983.
M.A. Diniz-Ehrhardt, Z. Dostál, M.A. Gomes-Ruggiero, J.M. Martýnez, and S.A. Santos, "Nonmonotone strategy for minimization of quadratics with simple constraints," Appl. Math., vol. 46, no. 5, pp. 321–338, 2001.
M.A. Diniz-Ehrhardt, M.A. Gomes-Ruggiero, and S.A. Santos, "Numerical analysis of the leaving-face crite-rion in bound-constrained quadratic minimization," Optimization Methods and Software, vol. 15, pp. 45–66, 2001.
Z. Dostál, "Box constrained quadratic programming with proportioning and projections," SIAM J. Optimiza-tion, vol. 7, no. 3, pp. 871–887, 1997.
Z. Dostál, "Inexact solution of auxiliary problems in Polyak type algorithms," Acta Univ. Palacki. Olomuc., Fac. Rer. Nat., Mathematica, vol. 38, pp. 25–30, 1999.
Z. Dostál, "A proportioning based algorithm for bound constrained quadratic programming with the rate of convergence," Numerical Algorithms, vol. 34, nos. 2–4, pp. 293–302, 2003.
Z. Dostál, A. Friedlander, and S.A. Santos, "Adaptive precision control in quadratic programming with simple bounds and/or equalities," in High Performance Software for Non-linear Optimization, R. De Leone, A. Murli, P. M. Pardalos, and G. Toraldo (Eds.), Kluwer, Applied Optimization, 1998, vol. 24, pp. 161–173.
Z. Dostál, A. Friedlander, and S.A. Santos, "Augmented Lagrangians with adaptive precision control for quadratic programming with simple bounds and equality constraints," SIAM J. Optimization, vol. 13, no. 4, pp. 1120–1140, 2003.
Z. Dostál, F.A.M. Gomes, and S.A. Santos, "Duality based domain decomposition with natural coarse space for variational inequalities," J. Comput. Appl. Math., vol. 126, nos. 1–2, pp. 397–415, 2000.
Z. Dostál, F.A.M. Gomes, and S.A. Santos, "Solution of Contact problems by FETI domain decomposition with natural coarse space projection," Computer Meth. in Appl. Mech. and Engineering, vol. 190, nos. 13–14,pp. 1611–1627, 2000.
Z. Dostál and D. Horák, "Scalability and FETI based algorithm for large discretized variational inequalities," Math. and Comp. in Simulation, vol. 61, nos. 3–6, pp. 347–357, 2003.
Z. Dostál and D. Horák, " Scalable FETI with optimal dual penalty for variational inequalities," Num. Lin. Algebra and Applications, to appear.
Z. Dostál, D. Horák and D. Stefanica, "atA scalable FETI-DP algorithm for coercive variational inequalities," IMACS Journal Applied Numerical Mathematics, submitted.
C. Farhat, J. Mandel, and F.X. Roux, "Optimal convergence properties of the FETI domain decomposition method," Computer Methods in Applied Mechanics and Engineering vol. 115, pp. 365–385, 1994.
R. Fletcher, Practical Methods of Optimization. John Wiley & Sons, Chichester, 1997.
A. Friedlander and M. Martýnez, "On the maximization of a concave quadratic function with box constraints," SIAM J. Optimization, vol. 4, pp. 177–192, 1994.
A. Friedlander, J.M. Martýnez, and S.A. Santos, "A new trust region algorithm for bound constrained mini-mization," Applied Mathematics & Optimization, vol. 30, pp. 235–266, 1994.
A. Friedlander, J.M. Martýnez, and M. Raydan, "A new method for large scale box constrained quadratic minimization problems," Optimization Methods and Software, vol. 5, pp. 57–74, 1995.
A. Greenbaum, Iterative Methods for Solving Linear Systems. SIAM, Philaledelphia 1997.
Z.-Q. Luo and P. Tseng, "Error bounds and convergence analysis of feasible descent methods: A general approach," Annals of Operations Research, vol. 46, pp. 157–178, 1993.
J.J. Moré and G. Toraldo, "On the solution of large quadratic programming problems with bound constraints," SIAM J. Optim., vol. 1, pp. 93–113, 1991.
D.P. O'Leary, "A generalised conjugate gradient algorithm for solving a class of quadratic programming problems," Lin. Alg. Appl., vol. 34, pp. 371–399, 1980.
B.T. Polyak, "The conjugate gradient method in extremal problems," USSR Comput. Math. and Math. Phys., vol. 9, pp. 94–112, 1969.
J. Schöberl, "Solving the Signorini problem on the basis of domain decomposition techniques," Computing", vol. 60, no. 4, pp. 323–344, 1998.
J. Schöberl, "Efficient contact solvers based on domain decomposition techniques," Int. J. Computers & Mathematics with Applications, vol. 42, pp. 1217–1228, 2001.
A. van der Sluis and H.A. van der Vorst, "The rate of convergence of the conjugate gradients," Numer. Math., vol. 48, pp. 543–560, 1986.
S.J. Wright Primal-Dual Interior Point Methods. SIAM, Philadelphia, 1997.
E.K. Yang and J.W. Tolle, "A class of methods for solving large, convex quadratic programs subject to box constraints," Mathematical Programming, vol. 51, pp. 223–228, 1991.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dostal, Z., Schoberl, J. Minimizing Quadratic Functions Subject to Bound Constraints with the Rate of Convergence and Finite Termination. Computational Optimization and Applications 30, 23–43 (2005). https://doi.org/10.1023/B:COAP.0000049888.80264.25
Issue Date:
DOI: https://doi.org/10.1023/B:COAP.0000049888.80264.25