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:

\begin{subequations}\begin{gather}-500 \leqslant P_x\leqslant 500, \\ -500 \leqs...
...leqslant 999, \\ -999 \leqslant p_z\leqslant 999. \end{gather}\end{subequations}    

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

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

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

\begin{subequations}\begin{gather}0 \leqslant \theta \leqslant \pi, \\ 0 \leqsla...
...0 \leqslant \sigma_{\textrm{max,2}}\leqslant \pi. \end{gather}\end{subequations}    

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

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

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

The relax user manual (PDF), created 2016-10-28.