def test_minimize_poly():
assert minimize([x - 2 * y, x**2 + y**2 <= 1], x, y) == (-sqrt(5), {
x:
4 / sqrt(5) - sqrt(5),
y:
2 / sqrt(5)
})
def test_minimize_linear():
assert minimize([
-2 * x - 3 * y - 2 * z, 2 * x + y + z <= 4, x + 2 * y + z <= 7, z <= 5,
x >= 0, y >= 0, z >= 0
], x, y, z) == (-11, {
x: 0,
y: 3,
z: 1
})
assert minimize([
-2 * x - 3 * y - 2 * z, 2 * x + y + z <= 4, x + 2 * y + z <= 7, z <= 5,
x >= 0, y >= 0, z >= 0
], x, y, z) == (-11, {
x: 0,
y: 3,
z: 1
})
assert minimize([
-2 * x - 3 * y - 2 * z, 2 * x + y + z <= 4, x + 2 * y + z < 7, z <= 5,
x >= 0, y >= 0, z >= 0
], x, y, z) is None
assert minimize([
-2 * x - 3 * y - 4 * z, 3 * x + 2 * y + z <= 10,
2 * x + 5 * y + 3 * z <= 15, x >= 0, y >= 0, z >= 0
], x, y, z) == (-20, {
x: 0,
y: 0,
z: 5
})
assert maximize([
12 * x + 40 * y, x + y <= 15, x + 3 * y <= 36, x <= 10, x >= 0, y >= 0
], x, y) == (480, {
x: 0,
y: 12
})
assert minimize([
-2 * x - 3 * y - 4 * z,
Eq(3 * x + 2 * y + z, 10),
Eq(2 * x + 5 * y + 3 * z, 15), x >= 0, y >= 0, z >= 0
], x, y, z) == (Rational(-130, 7), {
x: Rational(15, 7),
y: 0,
z: Rational(25, 7)
})
def test_minimize_linear_2():
assert minimize([
2 * x + 3 * y - z, 1 <= x + y + z, x + y + z <= 2, 1 <= x - y + z,
x - y + z <= 2,
Eq(x - y - z, 3)
], x, y, z) == (3, {
x: 2,
y: -S.Half,
z: -S.Half
})
def test_minimize_poly():
assert minimize([x - 2*y, x**2 + y**2 <= 1], x, y)[0] == -sqrt(5)
def test_minimize_linear():
assert minimize([-2*x - 3*y - 2*z, 2*x + y + z <= 4,
x + 2*y + z <= 7, z <= 5, x >= 0,
y >= 0, z >= 0], x, y, z) == (-11, {x: 0, y: 3, z: 1})
assert minimize([-2*x - 3*y - 2*z, 2*x + y + z <= 4,
x + 2*y + z <= 7, z <= 5, x >= 0,
y >= 0, z >= 0], x, y, z) == (-11, {x: 0, y: 3, z: 1})
assert minimize([-2*x - 3*y - 2*z, 2*x + y + z <= 4,
x + 2*y + z < 7, z <= 5, x >= 0,
y >= 0, z >= 0], x, y, z) is None
assert minimize([-2*x - 3*y - 4*z, 3*x + 2*y + z <= 10,
2*x + 5*y + 3*z <= 15, x >= 0,
y >= 0, z >= 0], x, y, z) == (-20, {x: 0, y: 0, z: 5})
assert maximize([12*x + 40*y, x + y <= 15, x + 3*y <= 36,
x <= 10, x >= 0, y >= 0], x, y) == (480, {x: 0, y: 12})
assert minimize([-2*x - 3*y - 4*z, Eq(3*x + 2*y + z, 10),
Eq(2*x + 5*y + 3*z, 15), x >= 0, y >= 0, z >= 0],
x, y, z) == (Rational(-130, 7),
{x: Rational(15, 7), y: 0,
z: Rational(25, 7)})
assert maximize([2*x + y, y - x <= 1, x - 2*y <= 2,
x >= 0, y >= 0], x, y) == (oo, {x: nan, y: nan})
assert minimize([2*x + 3*y - z, 1 <= x + y + z,
x + y + z <= 2, 1 <= x - y + z,
x - y + z <= 2, Eq(x - y - z, 3)],
x, y, z) == (3, {x: 2, y: Rational(-1, 2), z: Rational(-1, 2)})
x1, x2, x3, x4, x5 = symbols('x1:6')
assert minimize([-2*x1 + 4*x2 + 7*x3 + x4 + 5*x5,
Eq(-x1 + x2 + 2*x3 + x4 + 2*x5, 7),
Eq(-x1 + 2*x2 + 3*x3 + x4 + x5, 6),
Eq(-x1 + x2 + x3 + 2*x4 + x5, 4),
x2 >= 0, x3 >= 0, x4 >= 0, x5 >= 0],
x1, x2, x3, x4, x5) == (19, {x1: -1, x2: 0,
x3: 1, x4: 0, x5: 2})
assert minimize([-x - y, x + 2*y <= 8, 3*x + 2*y <= 12,
x + 3*y >= 6, x >= 0, y >= 0],
x, y) == (-5, {x: 2, y: 3})
pytest.raises(InfeasibleProblem,
lambda: minimize([-x - y, x + 2*y <= 8, 3*x + 2*y <= 12,
x + 3*y >= 13, x >= 0, y >= 0],
x, y))
pytest.raises(InfeasibleProblem,
lambda: minimize([-x - y, x + 2*y <= 8, 3*x + 2*y <= 12,
x + 3*y >= 13, x >= 0, y >= 0], x, y))
assert minimize([-x - y, 2*x + y >= 4,
Eq(x + 2*y, 6), x >= 0, y >= 0],
x, y) == (-6, {x: 6, y: 0})
assert minimize([6*x + 3*y, x + y >= 1, 2*x - y >= 1,
3*y <= 2, x >= 0,
y >= 0], x, y) == (5, {x: Rational(2, 3),
y: Rational(1, 3)})
def test_minimize():
assert minimize(1) == (1, {})
assert minimize((x - 2)**2) == (0, {x: 2})
assert minimize((x - 2)**2, x) == (0, {x: 2})
assert minimize(1/x, x) == (-oo, {x: 0})
assert minimize(2*x**2 - 4*x + 5, x) == (3, {x: 1})
assert minimize([2*x**2 - 4*x + 5, x > 0], x) == (3, {x: 1})
assert minimize([2*x**2 - 4*x + 5, x > 1], x) is None
assert minimize([2*x**2 - 4*x + 5, x >= 2], x) == (5, {x: 2})
assert minimize([2*x**2 - 4*x + 5, x**2 <= 0], x) == (5, {x: 0})
assert minimize([x**2 - 1,
(x - 1)*(x - 2)*(x - 3)*(4 - x) >= 0]) == (0, {x: 1})
assert minimize([x**2, (x + 2)*(x - 1)*(1 - x) >= 0]) == (1, {x: 1})
assert minimize(sign(x), x) == (-1, {x: 0})
def test_minimize_analytic():
assert minimize(exp(x**2 + y**2), x, y) == (1, {x: 0, y: 0})
def test_minimize_linear():
assert minimize([
-2 * x - 3 * y - 2 * z, 2 * x + y + z <= 4, x + 2 * y + z <= 7, z <= 5,
x >= 0, y >= 0, z >= 0
], x, y, z) == (-11, {
x: 0,
y: 3,
z: 1
})
assert minimize([
-2 * x - 3 * y - 2 * z, 2 * x + y + z <= 4, x + 2 * y + z <= 7, z <= 5,
x >= 0, y >= 0, z >= 0
], x, y, z) == (-11, {
x: 0,
y: 3,
z: 1
})
assert minimize([
-2 * x - 3 * y - 2 * z, 2 * x + y + z <= 4, x + 2 * y + z < 7, z <= 5,
x >= 0, y >= 0, z >= 0
], x, y, z) is None
assert minimize([
-2 * x - 3 * y - 4 * z, 3 * x + 2 * y + z <= 10,
2 * x + 5 * y + 3 * z <= 15, x >= 0, y >= 0, z >= 0
], x, y, z) == (-20, {
x: 0,
y: 0,
z: 5
})
assert maximize([
12 * x + 40 * y, x + y <= 15, x + 3 * y <= 36, x <= 10, x >= 0, y >= 0
], x, y) == (480, {
x: 0,
y: 12
})
assert minimize([
-2 * x - 3 * y - 4 * z,
Eq(3 * x + 2 * y + z, 10),
Eq(2 * x + 5 * y + 3 * z, 15), x >= 0, y >= 0, z >= 0
], x, y, z) == (Rational(-130, 7), {
x: Rational(15, 7),
y: 0,
z: Rational(25, 7)
})
assert maximize([2 * x + y, y - x <= 1, x - 2 * y <= 2, x >= 0, y >= 0], x,
y) == (oo, {
x: nan,
y: nan
})
assert minimize([
2 * x + 3 * y - z, 1 <= x + y + z, x + y + z <= 2, 1 <= x - y + z,
x - y + z <= 2,
Eq(x - y - z, 3)
], x, y, z) == (3, {
x: 2,
y: Rational(-1, 2),
z: Rational(-1, 2)
})
x1, x2, x3, x4, x5 = symbols('x1:6')
assert minimize([
-2 * x1 + 4 * x2 + 7 * x3 + x4 + 5 * x5,
Eq(-x1 + x2 + 2 * x3 + x4 + 2 * x5, 7),
Eq(-x1 + 2 * x2 + 3 * x3 + x4 + x5, 6),
Eq(-x1 + x2 + x3 + 2 * x4 + x5, 4), x2 >= 0, x3 >= 0, x4 >= 0, x5 >= 0
], x1, x2, x3, x4, x5) == (19, {
x1: -1,
x2: 0,
x3: 1,
x4: 0,
x5: 2
})
assert minimize([
-x - y, x + 2 * y <= 8, 3 * x + 2 * y <= 12, x + 3 * y >= 6, x >= 0,
y >= 0
], x, y) == (-5, {
x: 2,
y: 3
})
pytest.raises(
InfeasibleProblem, lambda: minimize([
-x - y, x + 2 * y <= 8, 3 * x + 2 * y <= 12, x + 3 * y >= 13, x >=
0, y >= 0
], x, y))
pytest.raises(
InfeasibleProblem, lambda: minimize([
-x - y, x + 2 * y <= 8, 3 * x + 2 * y <= 12, x + 3 * y >= 13, x >=
0, y >= 0
], x, y))
assert minimize([-x - y, 2 * x + y >= 4,
Eq(x + 2 * y, 6), x >= 0, y >= 0], x, y) == (-6, {
x: 6,
y: 0
})
assert minimize([
6 * x + 3 * y, x + y >= 1, 2 * x - y >= 1, 3 * y <= 2, x >= 0, y >= 0
], x, y) == (5, {
x: Rational(2, 3),
y: Rational(1, 3)
})
def test_minimize_analytic():
assert minimize(exp(x**2 + y**2), x, y) == (1, {x: 0, y: 0})
def test_minimize_poly():
assert minimize([x - 2 * y, x**2 + y**2 <= 1], x, y)[0] == -sqrt(5)
def test_minimize():
assert minimize(1) == (1, {})
assert minimize((x - 2)**2) == (0, {x: 2})
assert minimize((x - 2)**2, x) == (0, {x: 2})
assert minimize(1 / x, x) == (-oo, {x: 0})
assert minimize(2 * x**2 - 4 * x + 5, x) == (3, {x: 1})
assert minimize([2 * x**2 - 4 * x + 5, x > 0], x) == (3, {x: 1})
assert minimize([2 * x**2 - 4 * x + 5, x > 1], x) is None
assert minimize([2 * x**2 - 4 * x + 5, x >= 2], x) == (5, {x: 2})
assert minimize([2 * x**2 - 4 * x + 5, x**2 <= 0], x) == (5, {x: 0})
assert minimize([x**2 - 1,
(x - 1) * (x - 2) * (x - 3) * (4 - x) >= 0]) == (0, {
x: 1
})
assert minimize([x**2, (x + 2) * (x - 1) * (1 - x) >= 0]) == (1, {x: 1})
assert minimize(sign(x), x) == (-1, {x: 0})
