Function GMP::Instance::CreateDual(GMP, name)

# GMP::Instance::CreateDual

The function GMP::Instance::CreateDual generates a mathematical program that is the dual representation of the specified generated mathematical program. The generated mathematical program should have model type LP.

GMP::Instance::CreateDual(
GMP,            ! (input) a generated mathematical program
name            ! (input) a string expression
)


## Arguments

GMP

An element in the set AllGeneratedMathematicalPrograms.

name

A string that holds the name for the dual of the generated mathematical program.

## Return Value

A new element in the set AllGeneratedMathematicalPrograms with the name as specified by the name argument.

Note

• The name argument should be different from the name of the original generated mathematical program.

• If an element with name specified by the name argument is already present in the set AllGeneratedMathematicalPrograms the corresponding generated mathematical program will be replaced (or updated in case the same symbolic mathematical program is involved).

• The solution of a dual variable can be accessed through the .ShadowPrice suffix of the corresponding (primal) constraint.

• Before a generated mathematical program is dualized, AIMMS first transforms it temporary into a standard form using the following rules:

• Each column $$x_i$$ that is frozen to 0 is deleted.

• For each column $$x_i$$ with upper bound $$u_i$$, $$u_i \neq 0$$ and $$u_i < \infty$$, an extra row $$x_i \leq u_i$$ is added.

• For each column $$x_i$$ with lower bound $$l_i$$, $$l_i \neq 0$$ and $$l_i > -\infty$$, an extra row $$x_i \geq l_i$$ is added.

• Each ranged row $$l_j \leq a^T x \leq u_j$$ ($$l_j > -\infty$$ and $$u_j < \infty$$) is replaced by two rows $$l_j \leq a^T x$$ and $$a^T x \leq u_j$$.

• By using the suffix .ExtendedConstraint it is possible to refer to the rows that are added to create the standard form:

• The constraint v.ExtendedConstraint('DualUpperBound',i) is added for a variable v(i) with an upper bound unequal to 0 and inf.

• The constraint v.ExtendedConstraint('DualLowerBound',i) is added for a variable v(i) with a lower bound unequal to 0 and -inf.

• The constraints c.ExtendedConstraint('DualLowerBound',j) and c.ExtendedConstraint('DualUpperBound',j) replace a ranged constraint c(j).

The solution of these constraints can be accessed through the .ShadowPrice suffix, e.g., v.ExtendedConstraint('DualUpperBound',i).ShadowPrice.

• The objective variable for the dual mathematical program will become mp.ExtendedConstraint(DualObjective) and the objective constraint will be mp.ExtendedVariable(DualDefinition), where mp denotes the symbolic mathematical program.

## Example

Assume that ‘PrimalModel’ is a mathematical program with the following declaration (in ams format):

Variable x1 {
Range: [0, 5];
}
Variable x2 {
Range: nonnegative;
}
Variable obj {
Definition: - 7 * x1 - 2 * x2;
}
Constraint c1 {
Definition: - x1 + 2 * x2 <= 4;
}
MathematicalProgram PrimalModel {
Objective: obj;
Direction: minimize;
Type: LP;
}


Then GMP::Instance::CreateDual will create a dual mathematical program with variables

name                                             lower  upper
c1                                                -inf      0
obj_definition                                    -inf    inf
x1.ExtendedConstraint('DualUpperBound')           -inf      0
PrimalModel.ExtendedConstraint('DualObjective')   -inf    inf


and constraints

x1:
- c1 + 7 * obj_definition + x1.ExtendedConstraint('DualUpperBound') >= 0 ;

x2:
+ 2 * c1 + 2 * obj_definition >= 0 ;

obj:
obj_definition = 1 ;

PrimalModel.ExtendedVariable('DualDefinition'):
- 4 * c1 - 5 * x1.ExtendedConstraint('DualUpperBound')
+ PrimalModel.ExtendedConstraint('DualObjective') = 0 ;


The function GMP::Instance::Generate. See Modifying an Extended Math Program Instance of the Language Reference for more details on extended suffixes.