Workshop on Biomechanics and Chemotaxis, Thu, 13 Dec, 2007
Speaker: Zdenek Dostal
Abstract
The talk will review our algorithms for quadratic programming problems and with applications in biomechanics.
First we describe an active set based algorithm [1] forthe solution of bound constrained quadratic programming problems.
The working set based algorithm combines the the conjugate gradient method to explore the face of the feasible region defined by the current iterate with the reduced gradient projection with the fixed steplength. The precision of approximate solutions of the auxiliary unconstrained problems is controlled by the structure of violation of the Karush-Kuhn-Tucker conditions at the active constraints. The algorithm has R-linear rate of convergence in terms of the spectral condition number of the Hessian matrix and the finite termination property preserved even for the dual degenerate problems. The algorithm is then combined with a variant of the augmented Lagrangian type algorithm for strictly convex quadratic programming problems with equality constraints to obtain and algorithm for the solution of the bound and equality constrained problems. The update rule for the penalty parameter is introduced that is related to the increase of the augmented Lagrangian. A qualitatively new feature of the algorithm is a bound on the feasibility error that is independent of conditioning of the constraints and is valid even for dependent constraints and the quadratic functions with semidefinite Hessian. The algorithm turned out to be a key ingredient in the development of scalable algorithms for the solution of variational inequalities (such as those describing contact problems in biomechanics) by FETI [3] and BETI [4] domain decomposition methods. We shall complete our talk by showing applications in biomechanics [5].
1] Z. Dostal and J. Schöberl, Minimizing quadratic functions subject to bound constraints with the rate of convergence and finite termination, Comput. Optimiz. and Applications.
[2] Z. Dostal, An optimal algorithm for bound and equality constrained quadratic programming problems with bounded spectrum, Computing 78 (2006) 311-328.
[3] Z. Dostal and D. Horak, Theoretically supported scalable FETI for numerical solution of variational inequalities, to appear in SIAM Journal on Numerical Analysis.
[4] J. Bouchala, Z. Dostal and M. Sadowska, Scalable BETI for Variational Inequalities, submitted to Computing, 2007
[5] Rasmussen, J., Vondrák, V., Damsgaard, M., Dostál, Z.: The algorithms of mathematical programming in muscle recruitment and muscle wrapping problems.
In III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering. Ed. C.A. Mota Soares et.al.Springer, 2006, CDROM, paper 2177
URL: www.ricam.oeaw.ac.at/specsem/ssqbm/participants/abstracts/index.php
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