Constructing linear constraints

In this section, you will learn about

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 jth variable does not appear in the ith constraint, the corresponding element of A, A_ij is set to zero.

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

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:

   Matrix A(10,12);
   Real a[] = {11,12,....,61,62,63,.....,101,102,...,1012}; 
   A << a;

Or, for small matrices you can enter the values as follows.

    Matrix A(3,2);
    A << 11 << 12
      << 21 << 22
      << 31 << 32;

Note: Matrices are stored by row.

Now, we show the necessary steps for creation of linear equations and inequalities.

Creating Linear Equations

As customary, linear equations are written as

$Ax = b.$

The LinearEquation constructor

   LinearEquation(const Matrix& A, const ColumnVector& rhs);

requires two parameters, a Matrix, which consists of the constraints, and a ColumnVector that contains the elements of the right-hand side.

Creating Linear Inequalities

In OPT++, the standard form for a linear inequality is

$Ax \ge b.$

The corresponding constructor is

   LinearInequality(const Matrix& A, const ColumnVector& rhs);

To define upper bounds on the constraints, such as

$Ax \le b,$

use the following constructor:

   LinearInequality(const Matrix& A, const ColumnVector& rhs, 
                    const bool rowFlag);

To create lower and upper bounds on the constraints, use

   LinearInequality(const Matrix& A, const ColumnVector& lower, 
                    const ColumnVector& upper);

which generates

$l \le Ax \le u.$

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

$l_i \le -1.0e10.$

Similarly, an infinite upper bound is specified by

$u_i \ge 1.0e10.$

Next Section: Constructing nonlinear constraints | Back to Main Page

Last revised July 13, 2006

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, last revised August 30, 2006.