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

hal.structure.identifierLaboratoire d'Analyse, Topologie, Probabilités [LATP]
dc.contributor.authorANGOT, Philippe
hal.structure.identifierTransferts, écoulements, fluides, énergétique [TREFLE]
dc.contributor.authorCALTAGIRONE, Jean-Paul
hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorFABRIE, Pierre
dc.date.accessioned2024-04-04T02:26:40Z
dc.date.available2024-04-04T02:26:40Z
dc.date.created2011-01-01
dc.date.issued2012
dc.identifier.issn0893-9659
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/189970
dc.description.abstractEnThe 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.
dc.language.isoen
dc.publisherElsevier
dc.subject.enConstrained optimization
dc.subject.enSaddle-point problems
dc.subject.enAugmented Lagrangian
dc.subject.enPenalty method
dc.subject.enSplitting prediction-correction scheme
dc.subject.enVector penalty-projection methods
dc.title.enA new fast method to compute saddle-points in constrained optimization and applications
dc.typeArticle de revue
dc.identifier.doi10.1016/j.aml.2011.08.015
dc.subject.halMathématiques [math]/Optimisation et contrôle [math.OC]
dc.subject.halMathématiques [math]/Analyse numérique [math.NA]
dc.subject.halMathématiques [math]/Equations aux dérivées partielles [math.AP]
dc.subject.halPhysique [physics]/Mécanique [physics]/Mécanique des fluides [physics.class-ph]
dc.subject.halSciences de l'ingénieur [physics]/Mécanique [physics.med-ph]/Mécanique des fluides [physics.class-ph]
dc.subject.halSciences de l'ingénieur [physics]/Milieux fluides et réactifs
bordeaux.journalApplied Mathematics Letters
bordeaux.page245-251
bordeaux.volume25
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.issue3
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
bordeaux.peerReviewedoui
hal.identifierhal-00626163
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-00626163v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Applied%20Mathematics%20Letters&rft.date=2012&rft.volume=25&rft.issue=3&rft.spage=245-251&rft.epage=245-251&rft.eissn=0893-9659&rft.issn=0893-9659&rft.au=ANGOT,%20Philippe&CALTAGIRONE,%20Jean-Paul&FABRIE,%20Pierre&rft.genre=article


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