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| HAL: hal-00626163, version 1 |
| DOI: 10.1016/j.aml.2011.08.015 |
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| Applied Mathematics Letters 25, 3 (2012) 245-251 |
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| A new fast method to compute saddle-points in constrained optimization and applications |
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Philippe Angot 1Jean-Paul Caltagirone 2 |
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| (2012) |
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| 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. |
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| 1: | Laboratoire d'Analyse, Topologie, Probabilités (LATP) |
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CNRS : UMR6632 – Université de Provence - Aix-Marseille I – Université Paul Cézanne - Aix-Marseille III |
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| 2: | Transferts, écoulements, fluides, énergétique (TREFLE) |
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CNRS : UMR8508 – Université Sciences et Technologies - Bordeaux I – École Nationale Supérieure de Chimie et de Physique de Bordeaux (ENSCPB) – Arts et Métiers ParisTech |
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| 3: | Institut de Mathématiques de Bordeaux (IMB) |
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CNRS : UMR5251 – Université Sciences et Technologies - Bordeaux I – Université Victor Segalen - Bordeaux II |
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| Subject | : | Mathematics/Optimization and Control Mathematics/Numerical Analysis Mathematics/Analysis of PDEs Physics/Mechanics/Mechanics of the fluids Engineering Sciences/Mechanics/Fluids mechanics Engineering Sciences/Reactive fluid environment |
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| Constrained optimization – Saddle-point problems – Augmented Lagrangian – Penalty method – Splitting prediction-correction scheme – Vector penalty-projection methods |
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| hal-00626163, version 1 | |
| http://hal.archives-ouvertes.fr/hal-00626163 | |
| oai:hal.archives-ouvertes.fr:hal-00626163 | |
| From: Philippe Angot | |
| Submitted on: Friday, 23 September 2011 16:20:59 | |
| Updated on: Friday, 11 November 2011 14:43:37 | |
