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 \le...
...eqslant 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 \leqs...
...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 2019-03-08.