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\begin{center} {\Large LINEAR INEQUALITIES} \end{center}
\begin{center} Solving sets of linear inequalities and equations \\
and linear/integer optimization for small problems \end{center}
\begin{center} Version 2.0 May 1991\end{center}
\begin{center} Herbert Melenk \\ Konrad-Zuse-Zentrum fuer
Informationstechnik \\
Heilbronner Str. 10 \\ D1000 Berlin 31 \\ Federal Republic of Germany \\
melenk@sc.zib-berlin.de \end{center}
\section{Introduction}
This package solves sets of linear inequalities with real valued numerical
coefficients by applying the method of Fourier and Motzkin, as described
by G.B. Dantzig in his famous work {\em Linear Programming and Extensions}.
As input a system of linear inequalities (algebraic expressions composed
with the relational operators $<=$ and $>=$) and equations are accepted,
together with an optional specification, in which sequence the variables
are to be eliminated and - if there is the freedom - which variable should
take an extremal value in its final interval.
\section{The Algorithm}
The set of input formulas is treated in the following steps:
\begin{itemize}
\item The variables of the system are determined (together with a general
analysis); variables, which are not mentioned explicitly during the
call, are placed in front of the variable list; the following steps
then are controlled by the variable list.
\item Each equation (if any) is used to substitute one variable in the whole
system; if here an inconsistency occurs, the procedure is stopped with
an error message.
\item The remaining variables then are eliminated from the system of
inequalities one after the other by the technique of Fourier/Motzkin:
a system for the variables ${x_1,\ldots ,x_n}$ is reordered in the form
\[\begin{array}{cc}
L_1(x_2,\ldots ,x_n) \ge x_1 & \\
\ldots & \\
L_k(x_2,\ldots ,x_n) \ge x_1 & \\
& x_1 \ge R_1(x_2,\ldots ,x_n) \\
& \ldots \\
& x_1 \ge R_m(x_2,\ldots ,x_n)
\end{array}\]
and a new system in ${x_2,\ldots ,x_n}$
\[\begin{array}{c}
L_1 \ge R_1, \ldots , L_1 \ge R_m, \\
\ldots \\
L_k \ge R_1, \ldots , L_k \ge R_m \\
\end{array}
\]
is established and processed in a recursive manner.
\item When the last variable is encountered, the Li and Ri are numbers
and they establish an interval; if that interval is not empty, an
appropriate value for xn can be selected;
\item During returning from the recursion, the values found already are
substituted into the local Li and Ri, making them numerical values
too and allowing a selection in a non empty interval;
\item Finally the remaining variables are computed from substitutions
into the equations.
\end{itemize}
This procedure is generally valid; its limitation lies in the fact, that
the number of inequalities grows with each elimination step, if a variable
occurs with both signs several times. In order to limit this effect as
much as possible, numerical limits are collected as early as possible (by
taking the min resp. max value) and the inequalities are converted into a
canonical form (which actually is {\em homogeneous integer polynomial} $\ge$
{\em number}), such that multiplicities can be determined and eliminated.
The internal operation is based completely on REDUCE standard quotients and
standard forms.
If some or all variables are restricted to integer values, the
branch-and-bound method is used as a macro algorithm:
\begin{itemize}
\item First a solution of the system is determined without integer
restriction.
\item If the solution contains non integer values $x_0$ for an integer
variable $x$, new subproblems are generated with the additional
restrictions $x<=[x_0]$ resp. $x>=[x_0]+1$, where $[]$ denotes the next
integer below (floor value). With more than one integer variable this
leads to a tree of subproblems.
\item Among all feasable solutions the best one is selected.
\end{itemize}
\section{Algebraic Interface}
When the package is loaded, the operators $>=$ and $<=$ are declared as
operators, such that they can be used in algebraic expressions. The
operator for the solution of a linear system is:
\begin{center}
{\tt LININEQ($<$sys$>$ [,$<$vars$>$] [,RECORD=T] [,INT=$<$ivars$>$])}
\end{center}
\noindent
where $<$sys$>$ is the list of equations/inequalities either as explicit
list or as a value which evaluates to such a list, $<$vars$>$ (optional)
is a list of variables and/or expressions of the type $<$var$>$ $=$ min or
$<$var$>$ = max which means, that the corresponding variable is to be
taken at the corresponding edge of its limitating interval. Note,
that the last (or only) variable is the first one which gets a value
assigned; so this should be the target function in a linear
optimization context. If the keyword parameter RECORD is equated
with T, the intermediate inequalities are included in the result.
An equation with the lhs INT denotes a list of variables, which
are restricted to integer values. Note, that this does not automatically
include a restrition to poitive numbers: such limitations must
be given explicitly in the system.
The result is either an empty list, if the system is inconsistent,
or a list of equations assigning a numeric value (in the current
domain) to each variable. If RECORD=T was selected, each element
of the result itself is a list with the equation for the numerical
value for the variable, followed by the lower and upper bounds
of the corresponding interval. In most cases these bounds are
formal expressions in the other (following) variables, often in
the form of calls to ``max''or ``min'' If a variable is unbound to one
side, an {\tt INF} or {\tt -INF} appears in the corresponding place for
``infinity''.
If the switch {\tt PRLININEQ} is set, the intervals for the selection
of variable values are printed during the evaluation.
If the switch {\tt TRLININEQ} is set, a complete trace of the
elimination is printed; that may help in cases, where the result of
the process is not understood immediately; however, this printing may
result in a large volume.
By default {\tt PRLININEQ} and {\tt TRLININEQ} are off.
If the switch {\tt TRLININEQINT} is set, a restricted protocol for
integer problems is printed, which only shows the action of the
branch-and-bound method.
\noindent An example:
\begin{verbatim}
linineq({ 5x1 - 4x2 + 13x3 - 2x4 + x5 = 20,
x1 - x2 + 5x3 - x4 + x5 = 8,
x1 + 6x2 - 7x3 + x4 + 5x5 = z,
x1>=0,x2>=0,x3>=0,x4>=0,x5>=0}, {z=min});
\end{verbatim}
\noindent Result:
\begin{verbatim}
12 4 60
{X5=0,X4=0,X3=----,X2=---,X1=0,Z= - ----}
7 7 7
\end{verbatim}
\noindent The same with the tower of intervals:
\begin{verbatim}
linineq({ 5x1 - 4x2 + 13x3 - 2x4 + x5 = 20,
x1 - x2 + 5x3 - x4 + x5 = 8,
x1 + 6x2 - 7x3 + x4 + 5x5 = z,
x1>=0,x2>=0,x3>=0,x4>=0,x5>=0}, {z=min}, record=t);
\end{verbatim}
\noindent Result:
\begin{verbatim}
{{X5=0,
- X4 + 7*X3 - 6*X2 - X1 + Z
------------------------------,
5
- X4 + 7*X3 - 6*X2 - X1 + Z
------------------------------},
5
{X4=0,
32*X3 - 11*X2 + 4*X1 + Z - 40
-------------------------------,
6
32*X3 - 11*X2 + 4*X1 + Z - 40
-------------------------------},
6
12
{X3=----,
7
7*X2 - 20*X1 + Z + 32
-----------------------,
16
7*X2 - 20*X1 + Z + 32
-----------------------},
16
4
{X2=---,
7
20*X1 - Z - 32
MAX(12*X1 - Z - 8,----------------,0),
7
- 12*X1 + 3*Z + 32
---------------------},
11
7*Z + 60 2*Z + 36 3*Z + 32
{X1=0,0,MIN(----------,----------,----------)},
72 19 12
- 60 - 60
{Z=-------,-------,INF}}
7 7
\end{verbatim}
Remark: The operators $>$ and $<$ currently are not handled by the
system, because in many cases there is no algebraic solution for extremal
problems with these operators. For example ${x>0}$ has no algebraic solution.
\section{Symbolic Interface}
The function {\tt linineq0} offers an interface to the algorithm for symbolic
mode operation:
\begin{center} {\tt linineq0(prob,vars,dir,rec)} \end{center}
where
\begin{description}
\item[prob] list (untagged) of linear algebraic expressions
$(e_1 e_2 \ldots e_n)$ which do NOT contain any relational
operator; the problem to be solved then is
$e_1 \ge 0, \ldots , e_n \ge 0$
\item[vars] list of variables (kernels); the variable list must be
complete; otherwise an error will occur.
\item[dir] {\tt nil} or a list of the form
{\tt ( ( var$_1$ . mm$_1$ ) (var$_2$ . mm$_2$ ) ... )}
where the $var_i$ are variables and the mmi are either
{\tt MAX} or {\tt MIN}.
\item[rec] if not {\tt nil}, the formal bounds for the intermediate
intervals are collected. They are available in the
fluid variable {\tt LININEQRECORD!*}: the value of this
variable is a list, where each element corresponds
to one element of {\bf vars} in the same sequence. Each
element is a list with the lower bound and the
upperbound as algebraic (prefix) expression.
\end{description}
The result is {\tt NIL} (if the system is inconsistent) or a list of the form
{\tt ((var$_1$ . val$_1$) (var$_2$ . val$_2$) ... (var$_n$ . val$_n$))}
where the val$_i$ are algebraic (prefix) expressions, that are integers
or quotients of integers if no domain mode is selected.
\end{document}