deb-python-pulp/doc/source/documentation/elastic.rst

Elastic Constraints

pulp

A constraint C(x) = c (equality may be replaced by ≤ or ≥) can be elasticized to the form

C(x) ∈ 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 ~LpConstraint.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 ∈ [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 ∈ [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) ∉ 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

Following are the methods of the return-value: