# 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 Following are the methods of the return-value: