The OPTLP Procedure

Data Input and Output

Subsections:

This subsection describes the PRIMALIN= and DUALIN= data sets required to warm start the primal and dual simplex algorithms, and the PRIMALOUT= and DUALOUT= output data sets.

Definitions of Variables in the PRIMALIN= Data Set

The PRIMALIN= data set has two required variables defined as follows:

_VAR_

specifies the name of the decision variable.

_STATUS_

specifies the status of the decision variable. It can take one of the following values:

B

basic variable

L

nonbasic variable at its lower bound

U

nonbasic variable at its upper bound

F

free variable

A

newly added variable in the modified LP model when using the BASIS=WARMSTART option

Note: The PRIMALIN= data set is created from the PRIMALOUT= data set that is obtained from a previous "normal" run of PROC OPTLP (one that uses only the DATA= data set as the input).

Definitions of Variables in the DUALIN= Data Set

The DUALIN= data set also has two required variables defined as follows:

_ROW_

specifies the name of the constraint.

_STATUS_

specifies the status of the slack variable for a given constraint. It can take one of the following values:

B

basic variable

L

nonbasic variable at its lower bound

U

nonbasic variable at its upper bound

F

free variable

A

newly added variable in the modified LP model when using the BASIS=WARMSTART option

Note: The DUALIN= data set is created from the DUALOUT= data set that is obtained from a previous "normal" run of PROC OPTLP (one that uses only the DATA= data set as the input).

Definitions of Variables in the PRIMALOUT= Data Set

The PRIMALOUT= data set contains the primal solution to the LP model; each observation corresponds to a variable of the LP problem. If the SAVE_ONLY_IF_OPTIMAL option is not specified, the PRIMALOUT= data set can contain an intermediate solution, if one is available. See Example 12.1 for an example of the PRIMALOUT= data set. The variables in the data set have the following names and meanings.

_OBJ_ID_

specifies the name of the objective function. This is particularly useful when there are multiple objective functions, in which case each objective function has a unique name.

Note: PROC OPTLP does not support simultaneous optimization of multiple objective functions in this release.

_RHS_ID_

specifies the name of the variable that contains the right-hand-side value of each constraint.

_VAR_

specifies the name of the decision variable.

_TYPE_

specifies the type of the decision variable. _TYPE_ can take one of the following values:

N

nonnegative

D

bounded (with both lower and upper bound)

F

free

X

fixed

O

other (with either lower or upper bound)

_OBJCOEF_

specifies the coefficient of the decision variable in the objective function.

_LBOUND_

specifies the lower bound on the decision variable.

_UBOUND_

specifies the upper bound on the decision variable.

_VALUE_

specifies the value of the decision variable.

_STATUS_

specifies the status of the decision variable. _STATUS_ can take one of the following values:

B

basic variable

L

nonbasic variable at its lower bound

U

nonbasic variable at its upper bound

F

free variable

A

superbasic variable (a nonbasic variable that has a value strictly between its bounds)

I

LP model infeasible (all decision variables have _STATUS_ equal to I)

For the interior point algorithm with IIS= OFF, _STATUS_ is blank.

The following values can appear only if IIS= ON. See the section Irreducible Infeasible Set for details.

I_L

the lower bound of the variable is needed for the IIS

I_U

the upper bound of the variable is needed for the IIS

I_F

both bounds of the variable needed for the IIS (the variable is fixed or has conflicting bounds)

_R_COST_

specifies the reduced cost of the decision variable, which is the amount by which the objective function is increased per unit increase in the decision variable. The reduced cost associated with the ith variable is the ith entry of the following vector:

\[ \left(\mathbf{c}^\mr {T} - \mathbf{c}^\mr {T}_ B \mathbf{B}^{-1}\mathbf{A}\right) \]

where $\mathbf{B} \in \mathbb {R}^{m \times m}$ denotes the basis (matrix composed of basic columns of the constraints matrix $\mathbf{A} \in \mathbb {R}^{m \times n}$), $\mathbf{c} \in \mathbb {R}^ n$ is the vector of objective function coefficients, and $\mathbf{c}_ B \in \mathbb {R}^ m$ is the vector of objective coefficients of the variables in the basis.

Definitions of Variables in the DUALOUT= Data Set

The DUALOUT= data set contains the dual solution to the LP model; each observation corresponds to a constraint of the LP problem. If the SAVE_ONLY_IF_OPTIMAL option is not specified, the PRIMALOUT= data set can contain an intermediate solution, if one is available. Information about the objective rows of the LP problems is not included. See Example 12.1 for an example of the DUALOUT= data set. The variables in the data set have the following names and meanings.

_OBJ_ID_

specifies the name of the objective function. This is particularly useful when there are multiple objective functions, in which case each objective function has a unique name.

Note: PROC OPTLP does not support simultaneous optimization of multiple objective functions in this release.

_RHS_ID_

specifies the name of the variable that contains the right-hand-side value of each constraint.

_ROW_

specifies the name of the constraint.

_TYPE_

specifies the type of the constraint. _TYPE_ can take one of the following values:

L

"less than or equals" constraint

E

equality constraint

G

"greater than or equals" constraint

R

ranged constraint (both "less than or equals" and "greater than or equals")

_RHS_

specifies the value of the right-hand side of the constraint. It takes a missing value for a ranged constraint.

_L_RHS_

specifies the lower bound of a ranged constraint. It takes a missing value for a non-ranged constraint.

_U_RHS_

specifies the upper bound of a ranged constraint. It takes a missing value for a non-ranged constraint.

_VALUE_

specifies the value of the dual variable associated with the constraint.

_STATUS_

specifies the status of the slack variable for the constraint. _STATUS_ can take one of the following values:

B

basic variable

L

nonbasic variable at its lower bound

U

nonbasic variable at its upper bound

F

free variable

A

superbasic variable (a nonbasic variable that has a value strictly between its bounds)

I

LP model infeasible (all decision variables have _STATUS_ equal to I)

The following values can appear only if option IIS= ON. See the section Irreducible Infeasible Set for details.

I_L

the "GE" ($\geq $) condition of the constraint is needed for the IIS

I_U

the "LE" ($\leq $) condition of the constraint is needed for the IIS

I_F

both conditions of the constraint are needed for the IIS (the constraint is an equality or a range constraint with conflicting bounds)

_ACTIVITY_

specifies the left-hand-side value of a constraint. In other words, the value of _ACTIVITY_ for the ith constraint would be equal to $\mathbf{a}_ i^\mr {T}\mathbf{x}$, where $\mathbf{a}_ i$ refers to the ith row of the constraints matrix and $\mathbf{x}$ denotes the vector of current decision variable values.