22070 articles – 15901 Notices  [english version]

hal-00626163, version 1

## A new fast method to compute saddle-points in constrained optimization and applications

Philippe Angot () 1, Jean-Paul Caltagirone () 2, Pierre Fabrie () 3

Applied Mathematics Letters 25, 3 (2012) 245-251

Résumé : The solution of the augmented Lagrangian related system $(A+r\,B^TB)\,\rv=f$ is a key ingredient of many iterative algorithms for the solution of saddle-point problems in constrained optimization with quasi-Newton methods. However, such problems are ill-conditioned when the penalty parameter $\eps=1/r>0$ tends to zero, whereas the error vanishes as $\cO(\eps)$. We present a new fast method based on a {\em splitting penalty scheme} to solve such problems with a judicious prediction-correction. We prove that, due to the {\em adapted right-hand side}, the solution of the correction step only requires the approximation of operators independent on $\eps$, when $\eps$ is taken sufficiently small. Hence, the proposed method is all the cheaper as $\eps$ tends to zero. We apply the two-step scheme to efficiently solve the saddle-point problem with a penalty method. Indeed, that fully justifies the interest of the {\em vector penalty-projection methods} recently proposed in \cite{ACF08} to solve the unsteady incompressible Navier-Stokes equations, for which we give the stability result and some quasi-optimal error estimates. Moreover, the numerical experiments confirm both the theoretical analysis and the efficiency of the proposed method which produces a fast splitting solution to augmented Lagrangian or penalty problems, possibly used as a suitable preconditioner to the fully coupled system.

• 1 :  Laboratoire d'Analyse, Topologie, Probabilités (LATP)
• CNRS : UMR6632 – Université de Provence - Aix-Marseille I – Université Paul Cézanne - Aix-Marseille III
• 2 :  Transferts, écoulements, fluides, énergétique (TREFLE)
• CNRS : UMR8508 – Université Sciences et Technologies - Bordeaux I – École Nationale Supérieure de Chimie et de Physique de Bordeaux (ENSCPB) – Arts et Métiers ParisTech
• 3 :  Institut de Mathématiques de Bordeaux (IMB)
• CNRS : UMR5251 – Université Sciences et Technologies - Bordeaux I – Université Victor Segalen - Bordeaux II
• Domaine : Mathématiques/Optimisation et contrôle
Mathématiques/Analyse numérique
Mathématiques/Equations aux dérivées partielles
Physique/Mécanique/Mécanique des fluides
Sciences de l'ingénieur/Mécanique/Mécanique des fluides
Sciences de l'ingénieur/Milieux fluides et réactifs
• Mots-clés : Constrained optimization – Saddle-point problems – Augmented Lagrangian – Penalty method – Splitting prediction-correction scheme – Vector penalty-projection methods

• hal-00626163, version 1
• oai:hal.archives-ouvertes.fr:hal-00626163
• Contributeur :
• Soumis le : Vendredi 23 Septembre 2011, 16:20:59
• Dernière modification le : Vendredi 11 Novembre 2011, 14:43:37