
06046280
j
06046280
Angot, Philippe
Caltagirone, JeanPaul
Fabrie, Pierre
A new fast method to compute saddlepoints in constrained optimization and applications.
Appl. Math. Lett. 25, No. 3, 245251 (2012).
2012
Elsevier Science Ltd. (Pergamon), Oxford
EN
constrained optimization
saddlepoint problems
augmented Lagrangian
penalty method
splitting prediction
correction scheme
vector penaltyprojection methods
doi:10.1016/j.aml.2011.08.015
Summary: The solution of the augmented Lagrangian related system $(A+rB^TB)u=f$ is a key ingredient of many iterative algorithms for the solution of saddlepoint problems in constrained optimization with quasiNewton methods. However, such problems are illconditioned when the penalty parameter $\epsilon =1/r>0$ tends to zero, whereas the error vanishes as $\Cal O(\epsilon)$. We present a new fast method based on a {\it splitting penalty scheme} to solve such problems with a judicious predictioncorrection method. We prove that, due to the {\it adapted righthand side}, the solution of the correction step only requires the approximation of operators independent of $\epsilon $, when $\epsilon $ is taken sufficiently small. Hence, the proposed method is as cheaper as $\epsilon $ tends to zero. We apply the twostep scheme to efficiently solve the saddlepoint problem with a penalty method. Indeed, that fully justifies the interest of the {\it vector penaltyprojection methods} recently proposed by Angot et al. (2008) [19] to solve the unsteady incompressible NavierStokes equations, for which we give the stability result and some quasioptimal 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.