Linear constraints for the frame order models

Linear constraints are implemented for the frame order models using the log-barrier constraint algorithm in minfx, as this does not require the derivation of gradients.

The pivot point and average domain position parameter constraints in Ångstrom are:

These translation parameter restrictions are simply to stop the optimisation in the case of model failures. Converting these to the A⋅x $\geqslant$ b matrix notation required for the optimisation constraint algorithm, the constraints become

$$\begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 ... ...0 &-1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 &-1 \\ \end{pmatrix} \begin{pmatrix} P_x\\ P_y\\ P_z\\ p_x\\ p_y\\ p_z\\ \end{pmatrix} \geqslant \begin{pmatrix} -500 \\ -500 \\ -500 \\ -500 \\ -500 \\ -500 \\ -999 \\ -999 \\ -999 \\ -999 \\ -999 \\ -999 \\ \end{pmatrix}$$ (12.80)

For the order or motional amplitude parameters of the set $\mathfrak{S}$, the constraints used are

These reflect the range of validity of these parameters. Converting to the A⋅x $\geqslant$ b notation, the constraints are

$$\begin{pmatrix} 1 & 0 & 0 & 0 & 0 \\ -1& 0 & 0 & 0 & 0 \\ 0 & 1... ...0 & 0 & 0 &-1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 &-1 \\ \end{pmatrix} \begin{pmatrix} \theta \\ \theta_x \\ \theta_y \\ \sigma_{\textrm{max}}\\ \sigma_{\textrm{max,2}}\\ \end{pmatrix} \geqslant \begin{pmatrix} 0 \\ -\pi \\ 0 \\ -\pi \\ 0 \\ 0 \\ -\pi \\ 0 \\ -\pi \\ 0 \\ -\pi \\ \end{pmatrix}$$ (12.82)

The pseudo-elliptic cone model constraint θ_x $\geqslant$ θ_y is used to simplify the optimisation space by eliminating symmetry.