ANGOT, Philippe; CALTAGIRONE, Jean-Paul; FABRIE, Pierre

doi:10.1016/j.aml.2011.08.015

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ANGOT, Philippe
Laboratoire d'Analyse, Topologie, Probabilités [LATP]

CALTAGIRONE, Jean-Paul
Transferts, écoulements, fluides, énergétique [TREFLE]

FABRIE, Pierre
Institut de Mathématiques de Bordeaux [IMB]

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Applied Mathematics Letters. 2012, vol. 25, n° 3, p. 245-251

Elsevier

Résumé en anglais

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.< Réduire

Mots clés en anglais

Constrained optimization

Saddle-point problems

Augmented Lagrangian

Penalty method

Splitting prediction-correction scheme

Vector penalty-projection methods

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

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