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from sympy.core.numbers import Rational
from sympy.core.relational import Eq, Ne
from sympy.core.symbol import symbols
from sympy.core.sympify import sympify
from sympy.core.singleton import S
from sympy.core.random import random, choice
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.ntheory.generate import randprime
from sympy.matrices.dense import Matrix
from sympy.solvers.solveset import linear_eq_to_matrix
from sympy.solvers.simplex import (_lp as lp, _primal_dual,
UnboundedLPError, InfeasibleLPError, lpmin, lpmax,
_m, _abcd, _simplex, linprog)
from sympy.external.importtools import import_module
from sympy.testing.pytest import raises
from sympy.abc import x, y, z
np = import_module("numpy")
scipy = import_module("scipy")
def test_lp():
r1 = y + 2*z <= 3
r2 = -x - 3*z <= -2
r3 = 2*x + y + 7*z <= 5
constraints = [r1, r2, r3, x >= 0, y >= 0, z >= 0]
objective = -x - y - 5 * z
ans = optimum, argmax = lp(max, objective, constraints)
assert ans == lpmax(objective, constraints)
assert objective.subs(argmax) == optimum
for constr in constraints:
assert constr.subs(argmax) == True
r1 = x - y + 2*z <= 3
r2 = -x + 2*y - 3*z <= -2
r3 = 2*x + y - 7*z <= -5
constraints = [r1, r2, r3, x >= 0, y >= 0, z >= 0]
objective = -x - y - 5*z
ans = optimum, argmax = lp(max, objective, constraints)
assert ans == lpmax(objective, constraints)
assert objective.subs(argmax) == optimum
for constr in constraints:
assert constr.subs(argmax) == True
r1 = x - y + 2*z <= -4
r2 = -x + 2*y - 3*z <= 8
r3 = 2*x + y - 7*z <= 10
constraints = [r1, r2, r3, x >= 0, y >= 0, z >= 0]
const = 2
objective = -x-y-5*z+const # has constant term
ans = optimum, argmax = lp(max, objective, constraints)
assert ans == lpmax(objective, constraints)
assert objective.subs(argmax) == optimum
for constr in constraints:
assert constr.subs(argmax) == True
# Section 4 Problem 1 from
# http://web.tecnico.ulisboa.pt/mcasquilho/acad/or/ftp/FergusonUCLA_LP.pdf
# answer on page 55
v = x1, x2, x3, x4 = symbols('x1 x2 x3 x4')
r1 = x1 - x2 - 2*x3 - x4 <= 4
r2 = 2*x1 + x3 -4*x4 <= 2
r3 = -2*x1 + x2 + x4 <= 1
objective, constraints = x1 - 2*x2 - 3*x3 - x4, [r1, r2, r3] + [
i >= 0 for i in v]
ans = optimum, argmax = lp(max, objective, constraints)
assert ans == lpmax(objective, constraints)
assert ans == (4, {x1: 7, x2: 0, x3: 0, x4: 3})
# input contains Floats
r1 = x - y + 2.0*z <= -4
r2 = -x + 2*y - 3.0*z <= 8
r3 = 2*x + y - 7*z <= 10
constraints = [r1, r2, r3] + [i >= 0 for i in (x, y, z)]
objective = -x-y-5*z
optimum, argmax = lp(max, objective, constraints)
assert objective.subs(argmax) == optimum
for constr in constraints:
assert constr.subs(argmax) == True
# input contains non-float or non-Rational
r1 = x - y + sqrt(2) * z <= -4
r2 = -x + 2*y - 3*z <= 8
r3 = 2*x + y - 7*z <= 10
raises(TypeError, lambda: lp(max, -x-y-5*z, [r1, r2, r3]))
r1 = x >= 0
raises(UnboundedLPError, lambda: lp(max, x, [r1]))
r2 = x <= -1
raises(InfeasibleLPError, lambda: lp(max, x, [r1, r2]))
# strict inequalities are not allowed
r1 = x > 0
raises(TypeError, lambda: lp(max, x, [r1]))
# not equals not allowed
r1 = Ne(x, 0)
raises(TypeError, lambda: lp(max, x, [r1]))
def make_random_problem(nvar=2, num_constraints=2, sparsity=.1):
def rand():
if random() < sparsity:
return sympify(0)
int1, int2 = [randprime(0, 200) for _ in range(2)]
return Rational(int1, int2)*choice([-1, 1])
variables = symbols('x1:%s' % (nvar + 1))
constraints = [(sum(rand()*x for x in variables) <= rand())
for _ in range(num_constraints)]
objective = sum(rand() * x for x in variables)
return objective, constraints, variables
# equality
r1 = Eq(x, y)
r2 = Eq(y, z)
r3 = z <= 3
constraints = [r1, r2, r3]
objective = x
ans = optimum, argmax = lp(max, objective, constraints)
assert ans == lpmax(objective, constraints)
assert objective.subs(argmax) == optimum
for constr in constraints:
assert constr.subs(argmax) == True
def test_simplex():
L = [
[[1, 1], [-1, 1], [0, 1], [-1, 0]],
[5, 1, 2, -1],
[[1, 1]],
[-1]]
A, B, C, D = _abcd(_m(*L), list=False)
assert _simplex(A, B, -C, -D) == (-6, [3, 2], [1, 0, 0, 0])
assert _simplex(A, B, -C, -D, dual=True) == (-6,
[1, 0, 0, 0], [5, 0])
assert _simplex([[]],[],[[1]],[0]) == (0, [0], [])
# handling of Eq (or Eq-like x<=y, x>=y conditions)
assert lpmax(x - y, [x <= y + 2, x >= y + 2, x >= 0, y >= 0]
) == (2, {x: 2, y: 0})
assert lpmax(x - y, [x <= y + 2, Eq(x, y + 2), x >= 0, y >= 0]
) == (2, {x: 2, y: 0})
assert lpmax(x - y, [x <= y + 2, Eq(x, 2)]) == (2, {x: 2, y: 0})
assert lpmax(y, [Eq(y, 2)]) == (2, {y: 2})
# the conditions are equivalent to Eq(x, y + 2)
assert lpmin(y, [x <= y + 2, x >= y + 2, y >= 0]
) == (0, {x: 2, y: 0})
# equivalent to Eq(y, -2)
assert lpmax(y, [0 <= y + 2, 0 >= y + 2]) == (-2, {y: -2})
assert lpmax(y, [0 <= y + 2, 0 >= y + 2, y <= 0]
) == (-2, {y: -2})
# extra symbols symbols
assert lpmin(x, [y >= 1, x >= y]) == (1, {x: 1, y: 1})
assert lpmin(x, [y >= 1, x >= y + z, x >= 0, z >= 0]
) == (1, {x: 1, y: 1, z: 0})
# detect oscillation
# o1
v = x1, x2, x3, x4 = symbols('x1 x2 x3 x4')
raises(InfeasibleLPError, lambda: lpmin(
9*x2 - 8*x3 + 3*x4 + 6,
[5*x2 - 2*x3 <= 0,
-x1 - 8*x2 + 9*x3 <= -3,
10*x1 - x2+ 9*x4 <= -4] + [i >= 0 for i in v]))
# o2 - equations fed to lpmin are changed into a matrix
# system that doesn't oscillate and has the same solution
# as below
M = linear_eq_to_matrix
f = 5*x2 + x3 + 4*x4 - x1
L = 5*x2 + 2*x3 + 5*x4 - (x1 + 5)
cond = [L <= 0] + [Eq(3*x2 + x4, 2), Eq(-x1 + x3 + 2*x4, 1)]
c, d = M(f, v)
a, b = M(L, v)
aeq, beq = M(cond[1:], v)
ans = (S(9)/2, [0, S(1)/2, 0, S(1)/2])
assert linprog(c, a, b, aeq, beq, bounds=(0, 1)) == ans
lpans = lpmin(f, cond + [x1 >= 0, x1 <= 1,
x2 >= 0, x2 <= 1, x3 >= 0, x3 <= 1, x4 >= 0, x4 <= 1])
assert (lpans[0], list(lpans[1].values())) == ans
def test_lpmin_lpmax():
v = x1, x2, y1, y2 = symbols('x1 x2 y1 y2')
L = [[1, -1]], [1], [[1, 1]], [2]
a, b, c, d = [Matrix(i) for i in L]
m = Matrix([[a, b], [c, d]])
f, constr = _primal_dual(m)[0]
ans = lpmin(f, constr + [i >= 0 for i in v[:2]])
assert ans == (-1, {x1: 1, x2: 0}),ans
L = [[1, -1], [1, 1]], [1, 1], [[1, 1]], [2]
a, b, c, d = [Matrix(i) for i in L]
m = Matrix([[a, b], [c, d]])
f, constr = _primal_dual(m)[1]
ans = lpmax(f, constr + [i >= 0 for i in v[-2:]])
assert ans == (-1, {y1: 1, y2: 0})
def test_linprog():
for do in range(2):
if not do:
M = lambda a, b: linear_eq_to_matrix(a, b)
else:
# check matrices as list
M = lambda a, b: tuple([
i.tolist() for i in linear_eq_to_matrix(a, b)])
v = x, y, z = symbols('x1:4')
f = x + y - 2*z
c = M(f, v)[0]
ineq = [7*x + 4*y - 7*z <= 3,
3*x - y + 10*z <= 6,
x >= 0, y >= 0, z >= 0]
ab = M([i.lts - i.gts for i in ineq], v)
ans = (-S(6)/5, [0, 0, S(3)/5])
assert lpmin(f, ineq) == (ans[0], dict(zip(v, ans[1])))
assert linprog(c, *ab) == ans
f += 1
c = M(f, v)[0]
eq = [Eq(y - 9*x, 1)]
abeq = M([i.lhs - i.rhs for i in eq], v)
ans = (1 - S(2)/5, [0, 1, S(7)/10])
assert lpmin(f, ineq + eq) == (ans[0], dict(zip(v, ans[1])))
assert linprog(c, *ab, *abeq) == (ans[0] - 1, ans[1])
eq = [z - y <= S.Half]
abeq = M([i.lhs - i.rhs for i in eq], v)
ans = (1 - S(10)/9, [0, S(1)/9, S(11)/18])
assert lpmin(f, ineq + eq) == (ans[0], dict(zip(v, ans[1])))
assert linprog(c, *ab, *abeq) == (ans[0] - 1, ans[1])
bounds = [(0, None), (0, None), (None, S.Half)]
ans = (0, [0, 0, S.Half])
assert lpmin(f, ineq + [z <= S.Half]) == (
ans[0], dict(zip(v, ans[1])))
assert linprog(c, *ab, bounds=bounds) == (ans[0] - 1, ans[1])
assert linprog(c, *ab, bounds={v.index(z): bounds[-1]}
) == (ans[0] - 1, ans[1])
eq = [z - y <= S.Half]
assert linprog([[1]], [], [], bounds=(2, 3)) == (2, [2])
assert linprog([1], [], [], bounds=(2, 3)) == (2, [2])
assert linprog([1], bounds=(2, 3)) == (2, [2])
assert linprog([1, -1], [[1, 1]], [2], bounds={1:(None, None)}
) == (-2, [0, 2])
assert linprog([1, -1], [[1, 1]], [5], bounds={1:(3, None)}
) == (-5, [0, 5])
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