### 6.4 Optimization approach

The optimization approach as proposed in Wheatland et al. (2000) is closely related to the MHD relaxation approach. It shares with this method that a similar initial non-equilibrium state is iterated towards a NLFFF equilibrium. It solves a similar iterative equation as Equation (65)
but has additional terms, as explained below. The force-free and solenoidal conditions are solved by minimizing the functional
If the minimum of this functional at is attained then the NLFFF equations (2) – (4) are fulfilled. The functional is minimized by taking the functional derivative of Equation (68) with respect to an iteration parameter :
For vanishing surface terms the functional decreases monotonically if the magnetic field is iterated by
The first term in Equation (70) is identical with as defined in Equation (66).

A principal problem with the optimization and the MHD-relaxation approaches is that using the full magnetic field vector on the lower boundary does not guarantee the existence of a force-free configuration (see the consistency criteria in Section 4.6. Consequently, if fed with inconsistent boundary data, the codes cannot find a force-free configuration, but a finite residual Lorentz force and/or a finite divergence of the field remains in the 3D equilibrium. A way around this problem is to preprocess the measured photospheric data, as explained in Section 4.7. An alternative approach is that one allows deviations of the measured horizontal field vector and the corresponding field vector on the lower boundary of the computational box during the minimization of the functional (68). Wiegelmann and Inhester (2010) extended this functional by another term

where is a free parameter and the matrix contains information how reliable the data (mainly measurements of the horizontal photospheric field) are. With this approach inconsistencies in the measurement lead to a solution compatible with physical requirements (vanishing Lorentz force and divergence), leaving differences between and the bottom boundary field in regions where is low (and the measurement error high). Consequently, this approach takes measurement errors, missing data, and data inconsistencies into account. Further tests are necessary to investigate whether this approach or preprocessing, or a combination of both, is the most effective way to deal with noisy and inconsistent photospheric field measurements. This approach, as well as a variant of the Grad–Rubin method, have been developed in response to a joint study by DeRosa et al. (2009), where one of the main findings w as that force-free extrapolation codes should be able to incorporate measurement inconsistencies (see also Section 6.6).