def special_constraint(constraint, kind):
"""special constraints are the following inequalities
(1) x + y <= 1 | xy
(2) x1 + x2 + x3 <= 1 | x1x2 + x1x3 + x2x3
(3) x <= y | x - xy
(4) x = y | x + y - 2xy
(5) x + y >= 1 | 1- x- y + xy
(6) x + y = 1 | 1 - x- y + 2xy
Parameters
----------
constraint : sympy.core.expr.Expr or subclasses
constraint
kind : string
'lt', 'gt' or 'eq'
Returns
-------
sympy.core.expr.Expr, bool
constraint and Boolean, if it is a special constraint
"""
constraint_as_dict = Poly(constraint).as_expr().as_coefficients_dict()
constant = constraint_as_dict[1]
if constant == 0:
del constraint_as_dict[1]
length = len(constraint_as_dict)
else:
length = len(constraint_as_dict)
del constraint_as_dict[1]
sufficient_cond = [all(
coeff==1 or coeff==-1 for coeff in constraint_as_dict.values()),
constant in [0, -1],
length <= 4]
if not all(sufficient_cond):
return constraint, False
condition_eq_1_2 = [any([length == 3, length == 4]),
kind == 'lt',
all(coeff==1 for coeff in constraint_as_dict.values()),
constant==-1]
condition_eq_3_4 = [length == 2,
any([kind=='eq', kind=='lt']),
sum(constraint_as_dict.values())==0,
constant==0]
condition_eq_5_6 = [length == 3,
any([kind=='eq', kind == 'gt']),
all(coeff==1 for coeff in constraint_as_dict.values()),
constant==-1]
### x + y <= 1, x + y + z <= 1
if all(condition_eq_1_2):
var_combinations = sorted(
itermonomials(list(constraint.free_symbols), 2),
key=monomial_key('lex', list(constraint.free_symbols))
)
new_constraint = sum([
func for func in var_combinations
if (not func.is_integer) and Poly(func).is_multivariate
])
return new_constraint, True
### x <= y , x = y
elif all(condition_eq_3_4):
if kind == 'lt':
new_constraint = [var for var, coeff in constraint_as_dict.items()
if coeff==-1][0]
var_combinations = sorted(
itermonomials(list(constraint.free_symbols), 2),
key=monomial_key('lex', list(constraint.free_symbols))
)
new_constraint -= [term for term in var_combinations
if term.is_Mul][0]
return new_constraint, True
if kind == 'eq' :
var_combinations = sorted(
itermonomials(list(constraint.free_symbols), 2),
key=monomial_key('lex', list(constraint.free_symbols))
)
new_constraint = 0
for term in var_combinations:
if term.is_symbol:
new_constraint += term
elif term.is_Mul:
new_constraint -= 2*term
return new_constraint, True
### x + y >= 1, x + y = 1
if all(condition_eq_5_6):
var_combinations = sorted(
itermonomials(list(constraint.free_symbols), 2),
key=monomial_key('lex', list(constraint.free_symbols))
)
new_constraint = 0
for term in var_combinations:
if term.is_integer or term.is_Mul:
new_constraint += term
if term.is_Mul and kind=='eq':
new_constraint += term
if term.is_symbol and kind=='gt':
new_constraint -= term
if term.is_symbol and kind == 'eq':
new_constraint -= 2*term
return new_constraint, True
return constraint, False
