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OPT++

/** \page LinearConstraints Constructing linear constraints 

In this section, you will learn about 
<ul>
	<li> \ref Matrix,
	<li> \ref LinearEquations, and
	<li> \ref LinearInequalities.
</ul>
The two essential components of linear constraints, are the 
definitions of the constraints and the right-hand side.
Each row in a matrix denotes the coefficients of a linear
constraint.  If the \a jth variable does not appear in the
\a ith constraint, the corresponding element of A, A<sub>ij</sub> 
is set to zero. 

In the next section, we provide a brief tutorial on matrix initialization.

\section Matrix Initializing a Matrix in OPT++

Storing elements in a matrix is a simple task which can be accomplished in
various ways.  You can load the elements of a matrix from an array:
\code
   Matrix A(10,12);
   Real a[] = {11,12,....,61,62,63,.....,101,102,...,1012}; 
   A << a;
\endcode

Or, for small matrices you can enter the values as follows.
\code
    Matrix A(3,2);
    A << 11 << 12
      << 21 << 22
      << 31 << 32;
\endcode
<strong>Note:</strong> Matrices are stored by row.

Now, we show the necessary steps for creation of linear equations 
and inequalities.
\section LinearEquations Creating Linear Equations 
As customary, linear equations are written as \f[ Ax =  b. \f]

The LinearEquation constructor 
\code
   LinearEquation(const Matrix& A, const ColumnVector& rhs);
\endcode
requires two parameters, a Matrix, which consists of the constraints, and
a ColumnVector that contains the elements of the right-hand side.


\section LinearInequalities Creating Linear Inequalities
In OPT++, the standard form for a linear inequality is
\f[ Ax \ge  b. \f] 

The corresponding constructor is 
\code
   LinearInequality(const Matrix& A, const ColumnVector& rhs);
\endcode

To define upper bounds on the constraints, such as
\f[ Ax \le  b, \f] use the following constructor:
\code
   LinearInequality(const Matrix& A, const ColumnVector& rhs, 
	            const bool rowFlag);
\endcode

To create lower and upper bounds on the constraints,
use 
\code
   LinearInequality(const Matrix& A, const ColumnVector& lower, 
                    const ColumnVector& upper);
\endcode
which generates 
\f[l \le Ax \le  u. \f]

<p>
OPT++ does not support sparse constraints.  Therefore, a bound must be given 
for each constraint even if only a subset of the constraint have finite bounds.
An infinite lower bound is specified by
\f[l_i \le -1.0e10. \f]
Similarly, an infinite upper bound is specified by
\f[u_i \ge  1.0e10. \f]
</p>

<p> <a href="NonLinearConstraints.html">Next Section: Constructing nonlinear constraints
</a> | <a href="index.html">Back to Main Page</a> </p> 

Last revised <em> July 13, 2006</em>

*/