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

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Abstract : 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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https://hal.archives-ouvertes.fr/hal-00626163
Contributor : Philippe Angot <>
Submitted on : Friday, September 23, 2011 - 4:20:59 PM
Last modification on : Thursday, January 23, 2020 - 6:22:16 PM
Long-term archiving on: : Monday, December 5, 2016 - 2:19:37 AM

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Philippe Angot, Jean-Paul Caltagirone, Pierre Fabrie. A new fast method to compute saddle-points in constrained optimization and applications. Applied Mathematics Letters, Elsevier, 2012, 25 (3), pp.245-251. ⟨10.1016/j.aml.2011.08.015⟩. ⟨hal-00626163⟩

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