Basis descriptor. More...

#include <spxbasis.h>

Public Types

enum	Status { P_ON_LOWER = -4, P_ON_UPPER = -2, P_FREE = -1, P_FIXED = P_ON_UPPER + P_ON_LOWER, D_FREE = 1, D_ON_UPPER = 2, D_ON_LOWER = 4, D_ON_BOTH = D_ON_LOWER + D_ON_UPPER, D_UNDEFINED = 8 }
	Status of a variable. More...

Public Member Functions

int	nCols () const
	returns number of columns. More...

int	nRows () const
	returns number of rows. More...

int	dim () const
	returns dimension. More...

int	coDim () const
	returns codimension. More...

Status &	rowStatus (int i)

Status	rowStatus (int i) const
	returns status of row `i`. More...

const Status *	rowStatus (void) const
	returns the array of row Statuses. More...

Status &	colStatus (int i)

Status	colStatus (int i) const
	returns status of column `i`. More...

const Status *	colStatus (void) const
	returns the array of column Statuses. More...

Status &	status (int i)

Status	status (int i) const
	returns status of variable `i`. More...

const Status *	status (void) const
	returns the array of variable Statuses. More...

Status &	coStatus (int i)

Status	coStatus (int i) const
	returns status of covariable `i`. More...

const Status *	coStatus (void) const
	returns the array of covariable Statuses. More...

void	reSize (int rowDim, int colDim)
	resets dimensions. More...


void	dump () const
	Prints out status. More...

bool	isConsistent () const
	consistency check. More...


	Desc ()
	default constructor More...

	Desc (const SPxSolver &base)

	Desc (const Desc &old)
	copy constructor More...

Desc &	operator= (const Desc &rhs)
	assignment operator More...

Private Attributes

DataArray< Status >	rowstat
	status of rows. More...

DataArray< Status >	colstat
	status of columns. More...

DataArray< Status > *	stat
	basis' status. More...

DataArray< Status > *	costat
	cobasis' status. More...

Friends
class	SPxBasis

std::ostream &	operator<< (std::ostream &os, const Status &stat)

Detailed Description

Basis descriptor.

Definition at line 104 of file spxbasis.h.

Member Enumeration Documentation

◆ Status

enum Status

Status of a variable.

A basis is described by assigning a Status to all of the LP variables and covariables. This assignment is maintained by the basis Descriptor.

Variables and covariables (slackvariables) may have a primal or dual Status. The first type specifies that a variable is set on a primal bound, while the latter type indicates a dual variable to be set on a bound. If a row variable has a primal status, say P_ON_UPPER, this means that the upper bound of the inequality is set to be tight. Hence, in this case the upper bound must not be infinity.

Equivalently, if the status of a variable is dual, say D_ON_UPPER, it means that the dual variable corresponding to the upper bound inequality of this variable is set to 0.

For a column basis, primal Statuses correspond to nonbasic variables, while dual ones are basic. This is reversed for a row basis. We will now reveal in more detail the significance of variable Statuses.

Primal Variables

Consider a range inequality \(l_r \le a^T x \le u_r\) or bounds on a variable \(l_c \le x_c \le u_c\). The following table reveals what is implied if the corresponding variable or covariable is assigned to a primal Status:

\[ \begin{array}{lcl} l_c \le x_c \le u_c & \mbox{Status}(x_i) & l_r \le a^T x \le u_r \\ \hline x_c = u_c < \infty & \mbox{P\_ON\_UPPER} & a^T x = u_r < \infty \\ x_c = l_c > -\infty & \mbox{P\_ON\_LOWER} & a^T x = l_r > -\infty \\ -\infty < l_c = x_c = u_c < \infty & \mbox{P\_FIXED} & -\infty < l_r = a^T x = u_r < \infty \\ -\infty = l_i < x_i=0 < u_i = \infty & \mbox{P\_FREE} & -\infty = l_r < a^T x = 0 < u_r = \infty \\ \end{array} \]

Note that to determine whether a variable with Status stat is set to its upper bound, one can compute the test (-stat | -P_ON_UPPER). This will yield true even if the variable is fixed, i.e., sitting on both bounds at the same time.

Dual Variables

In principle for implementing the Simplex algorithm it would suffice to use only one dual Status. However, for performance reasons it is advisable to introduce various dual status types, reflecting the structure of the bounds. Given an upper bound \(u\) and a lower bound \(l\) of a constraint or variable, the following table indicates the setting of the dual Status of this variable.

\[ \begin{array}{cl} l \le ... \le u & \mbox{Status} \\ \hline -\infty < l \ne u < \infty & \mbox{D\_ON\_BOTH} \\ -\infty < l \ne u = \infty & \mbox{D\_ON\_UPPER} \\ -\infty = l \ne u < \infty & \mbox{D\_ON\_LOWER} \\ -\infty < l = u < \infty & \mbox{D\_FREE} \\ -\infty = l \ne u = \infty & \mbox{D\_UNDEFINED} \\ \end{array} \]

Note that unbounded primal variables are reflected by an D_UNDEFINED dual variable, since no reduced costs exist for them. To facilitate the assignment of dual Statuses, class SPxBasis provides methods dualStatus(), dualColStatus() and dualRowStatus)().

Enumerator
P_ON_LOWER	primal variable is set to its lower bound
P_ON_UPPER	primal variable is set to its upper bound
P_FREE	primal variable is left free, but unset
P_FIXED	primal variable is fixed to both bounds
D_FREE	dual variable is left free, but unset
D_ON_UPPER	dual variable is set to its upper bound
D_ON_LOWER	dual variable is set to its lower bound
D_ON_BOTH	dual variable has two bounds
D_UNDEFINED	primal or dual variable is undefined