Procedure GMP::Linearization::Add(GMP1, GMP2, solution, constraintSet, deviationsPermitted, penaltyMultipliers, linNo, jacTol)

GMP::Linearization::Add

The procedure GMP::Linearization::Add adds a linearization row to a generated mathematical program (GMP1) with respect to a solution (column level values and row marginals) of a second generated mathematical program (GMP2) for each row in a set of nonlinear constraints of that second generated mathematical program.

GMP::Linearization::Add(
     GMP1,                  ! (input) a generated mathematical program
     GMP2,                  ! (input) a generated mathematical program
     solution,              ! (input) a solution
     constraintSet,         ! (input) a set of nonlinear constraints
     deviationsPermitted,   ! (input) a binary parameter
     penaltyMultipliers,    ! (input) a double parameter
     linNo,                 ! (input) a linearization number
     [jacTol]               ! (optional) the Jacobian tolerance
     )

Arguments

GMP1

An element in AllGeneratedMathematicalPrograms.

GMP2

An element in AllGeneratedMathematicalPrograms.

solution

An integer scalar reference to a solution in the solution repository of GMP2.

constraintSet

A subset of AllNonLinearConstraints.

deviationsPermitted

A binary parameter over AllNonLinearConstraints indicating whether deviations are permitted for these linearizations. If yes, a new column will also be added to GMP1 with an objective coefficient equal to the double value given in penaltyMultiplier multiplied with the row marginal of the row in solution.

penaltyMultipliers

A double parameter over AllNonLinearConstraints representing the penalty multiplier that will be used if deviationsPermitted indicates that a deviation is permitted for the linearization.

linNo

An integer scalar reference to the rows and columns of the linearization.

jacTol

The Jacobian tolerance; if the Jacobian value (in absolute sense) of a column in a nonlinear row is smaller than this value, the column will not be added to the linearization of that row. The default is 1e-5.

Return Value

The procedure returns 1 on success, or 0 otherwise.

Note

• This procedure fails if one of the constraints is ranged.

• Rows and columns added before for linNo will be deleted first.

• This procedure will fail if GMP2 contains a column that is not part of GMP1. A column that is part of GMP1 but not of GMP2 will be ignored, i.e., no coefficient for that column will be added to the linearizations.

• The formula for the linearization of a scalar nonlinear inequality \(g(x,y) \leq b_j\) around the point \((x,y) = (x^0,y^0)\) is as follows.

  \[\begin{split}g(x^0,y^0) + \bigtriangledown g(x^0,y^0)^T \begin{bmatrix} x - x^0 \\ y - y^0 \end{bmatrix} - z_j \leq b_j\end{split}\]

  where \(z_j \geq 0\) is the extra column that is added if a deviation is permitted.

• For a scalar nonlinear equality \(g(x,y) = b_j\) the sense of the linearization depends on the shadow price of the equality in the solution. The sense will be ‘\(\leq\)‘ if the shadow price is negative and the optimization direction is minimization, or if the shadow price is positive and the optimization direction is maximization. The sense will be ‘\(\geq\)‘ if the shadow price is positive and the optimization direction is minimization, or if the shadow price is negative and the optimization direction is maximization.

• By using the suffixes .ExtendedConstraint and .ExtendedVariable it is possible to refer to the rows and columns that are added by GMP::Linearization::Add:

  • Constraint c.ExtendedConstraint('Linearizationk',j) is added for each nonlinear constraint c(j).

  • Variable c.ExtendedVariable('Linearizationk',j) is added for each nonlinear constraint c(j) if a deviation is permitted for constraint c(j).

  Here \(k\) denotes the value of the argument linNo.

See also

The routines GMP::Linearization::AddSingle and GMP::Linearization::Delete. See Modifying an Extended Math Program Instance of the Language Reference for more details on extended suffixes.