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

The solution of the augmented Lagrangian related system ( A + r B T B ) u = 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 ε = 1 / r > 0 tends to zero, whereas the error vanishes as O ( ε ) . We present a new fast method based on a splitting penalty scheme to solve such problems with a judicious prediction–correction method. We prove that, due to the adapted right-hand side, the solution of the correction step only requires the approximation of operators independent of ε , when ε is taken sufficiently small. Hence, the proposed method is as cheaper as ε 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 vector penalty-projection methods recently proposed by Angot et al. (2008) [19] 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.