Elastic Constraints

A constraint \(C(x) = c\) (equality may be replaced by \(\le\) or \(\ge\)) can be elasticized to the form

\[C(x) \in D\]

where \(D\) denotes some interval containing the value \(c\).

Define the constraint in two steps:

  1. instantiate constraint (subclass of LpConstraint) with target \(c\).

  2. call its makeElasticSubProblem() method which returns an object of type FixedElasticSubProblem (subclass of LpProblem) - its objective is the minimization of the distance of \(C(x)\) from \(D\).

constraint = LpConstraint(..., rhs = c)
elasticProblem = constraint.makeElasticSubProblem(
                       penalty = <penalty_value>,
                       proportionFreeBound = <freebound_value>,
                       proportionFreeBoundList = <freebound_list_value>,
                       )
where:

  • <penalty_value> is a real number

  • <freebound_value> \(a \in [0,1]\) specifies a symmetric target interval \(D = (c(1-a),c(1+a))\) about \(c\)

  • <freebound_list_value> = [a,b], a list of proportions \(a, b \in [0,1]\) specifying an asymmetric target interval \(D = (c(1-a),c(1+b))\) about \(c\)

The penalty applies to the constraint at points \(x\) where \(C(x) \not \in D\). The magnitude of <penalty_value> can be assessed by examining the final objective function in the .lp file written by LpProblem.writeLP().

Example:

>>> constraint_1 = LpConstraint('ex_1',sense=1,rhs=200)
>>> elasticProblem_1 = constraint_1.makeElasticSubproblem(penalty=1, proportionFreeBound = 0.01)
>>> constraint_2 = LpConstraint('ex_2',sense=0,rhs=500)
>>> elasticProblem_2 = constraint_2.makeElasticSubproblem(penalty=1,
proportionFreeBoundList = [0.02, 0.05])

  1. constraint_1 has a penalty-free target interval of 1% either side of the rhs value, 200

  2. constraint_2 has a penalty-free target interval of - 2% on left and 5% on the right side of the rhs value, 500

Freebound interval

Following are the methods of the return-value: