[Une méthode de contraintes successives pour résoudre les inégalités matricielles linéaires paramétriques]
Nous présentons une méthode de contraintes successives qui réduit le travail nécessaire pour résoudre les inégalités matricielles linéaires paramétriques de grande dimension. Une caractéristique importante de notre méthode est la décomposition hors ligne/en ligne du travail. Les calculs coûteux sont effectués à l'avance, hors ligne, pour nous permettre de résoudre le problème de manière très économique en ligne. La même méthode est aussi appliquée à l'approximation des solutions des problèmes d'optimisation SDP.
We present a successive constraint approach that makes it possible to cheaply solve large-scale linear matrix inequalities for a large number of parameter values. The efficiency of our method is made possible by an offline/online decomposition of the workload. Expensive computations are performed beforehand, in the offline stage, so that the problem can be solved very cheaply in the online stage. We also extend the method to approximate solutions to semidefinite programming problems.
Accepté le :
Publié le :
Robert O'Connor 1
@article{CRMATH_2017__355_6_723_0, author = {Robert O'Connor}, title = {A successive constraint approach to solving parameter-dependent linear matrix inequalities}, journal = {Comptes Rendus. Math\'ematique}, pages = {723--728}, publisher = {Elsevier}, volume = {355}, number = {6}, year = {2017}, doi = {10.1016/j.crma.2017.05.001}, language = {en}, }
Robert O'Connor. A successive constraint approach to solving parameter-dependent linear matrix inequalities. Comptes Rendus. Mathématique, Volume 355 (2017) no. 6, pp. 723-728. doi : 10.1016/j.crma.2017.05.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.05.001/
[1] Primal–dual interior-point methods for semidefinite programming: convergence rates, stability and numerical results, SIAM J. Optim., Volume 8 (1998) no. 3, pp. 746-768
[2] Control system analysis and synthesis via linear matrix inequalities, American Control Conference, June 1993, pp. 2147-2154
[3] A monotonic evaluation of lower bounds for inf–sup stability constants in the frame of reduced basis approximations, C. R. Acad. Sci. Paris, Ser. I, Volume 346 (2008) no. 23, pp. 1295-1300
[4] Analysis and synthesis of robust control systems via parameter-dependent Lyapunov functions, IEEE Trans. Autom. Control, Volume 41 (1996) no. 7, pp. 1041-1046
[5] Exploiting sparsity in semidefinite programming via matrix completion I: general framework, SIAM J. Optim., Volume 11 (2001) no. 3, pp. 647-674
[6] A spectral bundle method for semidefinite programming, SIAM J. Optim., Volume 10 (2000) no. 3, pp. 673-696
[7] A successive constraint linear optimization method for lower bounds of parametric coercivity and inf–sup stability constants, C. R. Acad. Sci. Paris, Ser. I, Volume 345 (2007) no. 8, pp. 473-478
[8] Lyapunov-based error bounds for the reduced-basis method, IFAC-PapersOnLine, Volume 49 (2016) no. 8, pp. 1-6
[9] Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations, Arch. Comput. Methods Eng., Volume 15 (2008) no. 3, pp. 229-275
[10] Solving large scale semidefinite programs via an iterative solver on the augmented systems, SIAM J. Optim., Volume 14 (2004) no. 3, pp. 670-698
[11] A tutorial on linear and bilinear matrix inequalities, J. Process Control, Volume 10 (2000) no. 4, pp. 363-385
[12] Semidefinite programming, SIAM Rev., Volume 38 (1996) no. 1, pp. 49-95
[13] A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations, Proceedings of the 16th AIAA Computational Fluid Dynamics Conference, 2003 (AIAA Paper 2003-3847)
Cité par Sources :
Commentaires - Politique