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

Autor: Jean-Paul Caltagirone, Philippe Angot, Pierre Fabrie
Přispěvatelé: Laboratoire d'Analyse, Topologie, Probabilités (LATP), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Transferts, écoulements, fluides, énergétique (TREFLE), Université Sciences et Technologies - Bordeaux 1 (UB)-École Nationale Supérieure de Chimie et de Physique de Bordeaux (ENSCPB)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Université Sciences et Technologies - Bordeaux 1-École Nationale Supérieure de Chimie et de Physique de Bordeaux (ENSCPB)-Centre National de la Recherche Scientifique (CNRS), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)
Rok vydání: 2012
Předmět:
Penalty method
010103 numerical & computational mathematics
Splitting prediction-correction scheme
01 natural sciences
[SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph]
\MSC[2010] 49K35
65F05
65F08
65F10
65F35
65K05
65K10
65K15
65N22
76D05
76D07
90C25
90C26
Saddle point
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]
[PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph]
0101 mathematics
Constrained optimization
Splitting prediction–correction scheme
Saddle
Mathematics
Augmented Lagrangian
Augmented Lagrangian method
Preconditioner
[SPI.FLUID]Engineering Sciences [physics]/Reactive fluid environment
Applied Mathematics
Mathematical analysis
Zero (complex analysis)
Vector penalty-projection methods
010101 applied mathematics
Compressibility
[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Saddle-point problems
[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Zdroj: Applied Mathematics Letters
Applied Mathematics Letters, 2012, 25 (3), pp.245-251. ⟨10.1016/j.aml.2011.08.015⟩
Applied Mathematics Letters, Elsevier, 2012, 25 (3), pp.245-251. ⟨10.1016/j.aml.2011.08.015⟩
ISSN: 0893-9659
DOI: 10.1016/j.aml.2011.08.015
Popis: International audience; 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.
Databáze: OpenAIRE
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