Linear Constraint Parameters

The following parameters can be passed to the APPSPACK::Constraints::Linear constructor.

"Lower" - (vector) The lower bounds
"Upper" - (vector) The upper bounds
"Scaling" - (vector) Scaling vector
"Inequality Matrix" - (matrix) The coefficent matrix of general linear inequality constraints.
"Inequality Upper" - (vector) The upper bounds on general linear inequality constraints.
"Inequality Lower" - (vector) The lower bounds on general linear inequality constraints.
"Equality Matrix" - (matrix) The coefficent matrix of linear equality constraints.
"Equality Bound" - (vector) Bound on equality constraints.
"Machine Epsilon" - (double) The tolerance to determine if a constraint is violated. Defaults to 1.0e-12.

Note:
When performing matrix operations such as null space basis generation, rank estimation, maximum feasible step computation, etc., it is important to allow for numerical round-off error. For example, when equality constraints are present, all search direction must lie within the null-space of the constraint normals. In exact arithmetic this equates to for all equality constraint normals where is the search direction. In the presence of numerical error, this requirement is replaced by $|a_i^T d_i| \le \epsilon$ .
"Display Linear Systems" (bool) - If true, equality and inequality constraint information will be printed explicitly for certain debug levels.

Input File Format

If using an input file, these parameters should be in the "Linear" sublist; see Using the APPSPACK Executables. (The use of "Bounds" as the sublist name is also allowed but deprecated.)

Consider the following example constraints:

$\begin{array}{ccccc} && x_1 & \leq & 5.7 \\ -0.6 & \leq & x_2 & & \\ 0 & \leq & x_3 & \leq & 1.3 \\ \end{array}$

$\begin{array}{ccc} & x_1 + x_2 + x_3 &\le .1 \\ -4.1 \le & 2x_1 - 9 x_3 & \\ & x_1 - x_2 + x_3 & = -5.2 \\ \end{array}$

The input file for the "Linear" parameters (see Step 2: Create an APPSPACK input file) is as follows:

@ "Linear"
"Lower" vector 3 DNE -0.6 0
"Upper" vector 3 5.7 DNE 1.3
"Scaling" vector 3 1 1 1
"Inequality Matrix" matrix 2 3
1  1  1
2  0  -9
"Inequality Upper" vector 2 .1    DNE
"Inequality Lower" vector 2 DNE  -4.1
"Equality Matrix" matrix 1 3
1  -1  1
"Equality Bound" vector 1 -5.2
@@

Note that we have explicitly defined "Scaling". This can only be definied implicity if the lower and upper bounds exist for every variable, in which case it is computed as

$s_i = u_i - \ell_i$

where $s_i$ represents the i-th entry of the scaling vector, $u_i$ represents the i-th upper bound, and $\ell_i$ represents the i-th lower bound. Here "DNE" is used to signify that the corresonding bound "Does Not Exist".

See also:: Step 2: Create an APPSPACK input file
Solver Parameters