\subsection{The \texttt{PLQ:} module}

The {\tt PLQ:} module is used to solve a general non-linear optimization problem with constraints using the Optex method.\cite{optex1,optex2,optex3,PIP2016}
The {\tt PLQ:} module performs a single iteration of the Optex method where a linearized optimization problem is solved with a quadratic constraint.
This quadratic constraint is of the form
$$
\sum_{i=1}^{\sl nvar} \omega_i \left( \Delta x_i^{(n)}\right)^2 \le \left( S^{(n)}\right)^2
$$
\noindent where $\omega_i$ is a weight defined after keyword \moc{CST-WEIGHT} in module {\tt GRAD:} and $\Delta x_i^{(n)}$ is a displacement for
$i$--th control variable at iteration $(n)$. The initial value of radius $S^{(1)}$ is defined after keyword \moc{OUT-STEP-LIM}.

\vskip 0.08cm

The gradients of the {\sl system characteristics} can be calculated with module {\tt GRAD:}. The {\tt PLQ:} module is also used to
define options and parameters for the different method to solve the optimization problem and to reduces the radius $S^{(n)}$ of
the quadratic constraint.

\vskip 0.08cm

The calling specifications are:

\begin{DataStructure}{Structure \moc{PLQ:}}
\dusa{OPTIM} \moc{:=} \moc{PLQ:} \dusa{OPTIM} \moc{::} \dstr{plq\_data}
\end{DataStructure}

\noindent where

\begin{ListeDeDescription}{mmmmmmmm}

\item[\dusa{OPTIM}] \texttt{character*12} name of the \dds{optimize} object ({\tt L\_OPTIMIZE} signature) containing the
optimization informations. Object \dusa{OPTIM} must appear on both LHS and RHS to be able to update the previous values.

\item[\dstr{plq\_data}] structure containing the data to the module \texttt{PLQ:} (see Sect.~\ref{sect:plq_data}).

\end{ListeDeDescription}
\vskip 0.2cm

\subsubsection{Data input for module \texttt{PLQ:}}\label{sect:plq_data}

\begin{DataStructure}{Structure \moc{plq\_data}}
$[$ \moc{EDIT} \dusa{iprint} $]$ \\
$[~\{$ \moc{MAXIMIZE} $|$ \moc{MINIMIZE} $\}~]$ \\
$[$ \moc{METHOD} \{ \moc{SIMPLEX} $|$ \moc{LEMKE} $|$ \moc{MAP} $|$ \moc{AUG-LAGRANG} $|$ \moc{PENAL-METH} \} $]$ \\
$[$ \moc{OUT-STEP-LIM} \dusa{sr} $]$ \\
$[$ \moc{OUT-STEP-EPS} \dusa{$\epsilon_{ext}$} $]~[$ \moc{INN-STEP-EPS} \dusa{$\epsilon_{inn}$} $]$ \\
$[$ \moc{CST-QUAD-EPS} \dusa{$\epsilon_{quad}$} $]$ \\
$[$ \moc{STEP-REDUCT} $\{$ \moc{HALF} $|$ \moc{PARABOLIC} $\}~]$ \\
$[$ \moc{WARNING-ONLY} $]$ \\
\moc{CALCUL-DX} $[$ \moc{NO-STORE-OLD} $]$ \\
$[$ \moc{COST-EXTRAP} {\tt >>} \dusa{ecost} {\tt <<} $]$ \\
$[$ \moc{OUT-CONV-TST} {\tt >>} \dusa{$l_{conv}$} {\tt <<} $]$ \\
;
\end{DataStructure}

\noindent where
\begin{ListeDeDescription}{mmmmmmmm}

\item[\moc{EDIT}] keyword used to set \dusa{iprint}.

\item[\dusa{iprint}] index used to control the printing in module.

\item[\moc{MAXIMIZE}] keyword used to specify that the optimization problem will be a maximization.

\item[\moc{MINIMIZE}] keyword used to specify that the optimization problem will be a minimization (default).

\item[\moc{METHOD}] keyword used to define the quasi-linear programming method. {\bf Note:} If the general Lemke method is
used, the quadratic constraint must be active. The strategy consists to proceed in two steps:
\begin{itemize}
\item At first step, the linear programming problem
(i. e., without the quadratic contraint) is solved and the control-variable displacement is computed. If this displacement is less
than the radius of the quadratic constraint, the step one solution is accepted and step two is not performed. If this displacement is greater
than the radius of the quadratic constraint, the step one solution is rejected and step two is performed. Step one can be
solved with the SIMPLEX method or with the linear LEMKE method.
\item At step two, the general LEMKE method is used to find the correct solution. The general Lemke method is based on a parametric linear
complementarity principle, as explained in Ref.~\citen{ferland}.
\end{itemize}

\item[\moc{SIMPLEX}] keyword used to specify that the SIMPLEX method will be used at step one and the general LEMKE method at step two.

\item[\moc{LEMKE}] keyword used to specify that the linear LEMKE method will be used at step one and the general LEMKE method at step two.

\item[\moc{MAP}] keyword used to specify that the MAP method will be used. The quadratic constraint is linearized and a converging sequence
of SIMPLEX calculations is performed.

\item[\moc{AUG-LAGRANG}] keyword used to specify that the augmented Lagrangian method will be used.

\item[\moc{PENAL-METH}] keyword used to specify that the penalty method will be used.

\item[\moc{OUT-STEP-LIM}] keyword used to set or reset the initial radius of the quadratic constraint (default value is \dusa{sr} $=1.0$ or the value
recovered from \dusa{OPTIM}).

\item[\dusa{sr}] initial radius of the quadratic constraint (real or double precision).

\item[\moc{OUT-STEP-EPS}] keyword used to set the tolerance of outer iteration convergence inside module {\tt PLQ:} (default value
is $1.0 \times 10^{-4}$).

\item[\dusa{$\epsilon_{ext}$}] tolerance value (real or double precision).

\item[\moc{INN-STEP-EPS}] keyword used to set the tolerance used within the SIMPLEX or LEMKE method (default value
is $1.0 \times 10^{-4}$).

\item[\dusa{$\epsilon_{inn}$}] tolerance value (real or double precision).

\item[\moc{CST-QUAD-EPS}] keyword to set the convergence parameter \dusa{epsilon4} for the radius of the quadratic constraint inside module {\tt GRAD:}.

\item[\dusa{$\epsilon_{quad}$}] tolerance for convergence of the radius of the quadratic constraint (real).

\item[\moc{STEP-REDUCT}] keyword used to define the method of the reduction of the outer step.

\item[\moc{HALF}] keyword used to specify that the step will be reduced by a factor of 2.

\item[\moc{PARABOLIC}] keyword used to specify that the step will be reduced with the parabolic method.

\item[\moc{WARNING-ONLY}] keyword used to specify that only a warning will be used when no valid previous decision vectors can
be recall in case of error of the mathematical programming.

\item[\moc{CALCUL-DX}] keyword used to specify that the new step will be calculated.

\item[\moc{NO-STORE-OLD}] keyword used to specify that the old value of decision variables and gradients will not be stored in
the {\tt L\_OPTIMIZE/'OLD-VALUE'} directory.

\item[\moc{COST-EXTRAP}] keyword used to calculate the extrapolated objective constant \dusa{ecost}.

\item[\dusa{ecost}] extrapolated objective constant.

\item[\moc{OUT-CONV-TST}] keyword used to calculate if the external convergence has been reached.

\item[\dusa{$l_{conv}$}] $=1$ means that external convergence has been reached; $=0$ otherwise.

\end{ListeDeDescription}
\clearpage