Abstract
The implementation of the recently proposed semi-monotonic augmented Lagrangian algorithm for the solution of large convex equality constrained quadratic programming problems is considered. It is proved that if the auxiliary problems are approximately solved by the conjugate gradient method, then the algorithm finds an approximate solution of the class of problems with uniformly bounded spectrum of the Hessian matrix at O(1) matrix–vector multiplications. If applied to the class of problems with the Hessian matrices that are in addition either sufficiently sparse or can be expressed as a product of such sparse matrices, then the cost of the solution is proportional to the dimension of the problems. Theoretical results are illustrated by numerical experiments.
Similar content being viewed by others
References
Axelsson, O.: Iterative Solution Methods.. Cambridge University Press, Cambridge (1994)
Bertsekas, D.P.: Nonlinear Optimization. Athena Scientific, Belmont (1999)
Dostál, Z.: On preconditioning and penalized matrices. Numer. Linear Algebra Appl. 6, 109–114 (1999)
Dostál, Z.: Semi-monotonic inexact augmented Lagrangians for quadratic programming with equality constraints. Optim. Methods Softw. 20(6), 715–727 (2005)
Dostál, Z.: Inexact semi-monotonic augmented Lagrangians with optimal feasibility convergence for quadratic programming with simple bounds and equality constraints. SIAM J. Numer. Anal. 43(1), 96–115 (2005)
Dostál, Z., Friedlander, A., Santos, S.A.: Adaptive precision control in quadratic programming with simple bounds and/or equalities. In: De Leone, R., Murli, A., Pardalos, P.M., Toraldo, G. (eds.) High Performance Software for Non-linear Optimization. Applied Optimization, vol. 24, pp. 161–173. Kluwer, Dordrecht (1998)
Dostál, Z., Friedlander, A., Santos, S.A.: Augmented Lagrangians with adaptive precision control for quadratic programming with equality constraints. Comput. Optim. Appl. 14, 37–53 (1999)
Dostál, Z., Friedlander, A., Santos, S.A., Alesawi, K.: Augmented Lagrangians with adaptive precision control for quadratic programming with equality constraints: corrigendum and addendum. Comput. Optim. Appl. 23(1), 127–133 (2002)
Elman, H.C., Golub, G.H.: Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J. Numer. Anal. 31(6), 1645–1661 (1994)
Glowinski, R., Le Tallec, P.: Augmented Lagrangians and Operator Splitting Methods. SIAM, Philadelphia (1989)
Golub, G.H., Van Loan, C.F.: Matrix Computations, 2nd. edn. John Hopkins University Press, Baltimore (1989)
Hager, W.W.: Analysis and implementation of a dual algorithm for constraint optimization. J. Optim. Theory Appl. 79, 37–71 (1993)
Hestenes, M.R.: Multiplier and gradient methods. J. Optim. Theory Appl. 4, 303–320 (1969)
Keller, C., Gould, N.I.M., Wathen, A.J.: Constraint preconditioning for indefinite linear systems. SIAM J. Matrix Anal. Appl. 21(4), 1300–1317 (2000)
Klawonn, A.: Block-triangular preconditioning for saddle pint problems with a penalty term. SIAM J. Sci. Comput. 19(1), 172–184 (1998)
Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (2000)
Powell, M.J.D.: A method for nonlinear constraints in minimization problems. In: Fletcher, R. (ed.) Optimization, pp. 283–298. Academic, New York (1969)
Saad, Y.: Iterative Methods for Sparse Systems. SIAM, Philadelphia (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research is supported by grants of the Ministry of Education No. S3086102, ET400300415 and MSM 6198910027.
Rights and permissions
About this article
Cite this article
Dostál, Z. An optimal algorithm for a class of equality constrained quadratic programming problems with bounded spectrum. Comput Optim Appl 38, 47–59 (2007). https://doi.org/10.1007/s10589-007-9036-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-007-9036-x