def test_ex82():
A = Matrix([
[0, 1, 0],
[0, 0, 1],
[1, -1, -1],
[3, 2, 2],
[-1, 0, 0],
[0, -1, 0],
[0, 0, -1],
])
b = Matrix(
[6, 9, 3, 24, 0, 0, 0]
)
x_0 = Matrix([8,0,0])
c = Matrix([1,1,-1])
I_x_0 = active_constraints(x_0, A, b)
assert I_x_0 == [3,5,6]
assert not is_contained(x_0, A, b)
v_feasible = determine_feasible_vertex_dimensions(A, b, I_x_0, pivot_rule_p=PivotRule.MAXIMAL(), pivot_rule_i=PivotRule.MAXIMAL())
assert v_feasible == Matrix([6, 0, 3])
x_p = Matrix([0,0,9])
assert is_contained(x_p, A, b)
B = next(iter(bases(x_p, A, b)))
res, v_star, opt_val, _ = simplex(A, b, c, x_p, set(B), pivot_rule_p=PivotRule.MAXIMAL(), pivot_rule_i=PivotRule.MAXIMAL())
assert v_star == Matrix([4, 6, 0])
def test_exam():
A = Matrix([
[1, 1, 1],
[1, -1, 1],
[0, 1, 1],
[0, -1, 1],
[-1, 0, 0],
[0, 0, -1]
])
b = Matrix([1, 1, 1, 1, 2, 0])
c = Matrix([0,4,6])
I = {0,1,5}
x_1 = Matrix([1,0,0])
simplex(A, b, c, x_1, I)
x_2 = Matrix([0,0,1])
I_2 = active_constraints(x_2, A, b)
B = next(iter(bases(x_2, A, b)))
simplex(A, b, c, x_2, B)
def test_ex85():
A = Matrix([
[1, 1],
[-1, 0],
[0, -1],
[1, 0]
])
b = Matrix([2, 0, 0, 1])
I = {0, 2}
assert not is_contained(sub_matrix(A, I)**-1*sub_matrix(b, I), A, b)
A_init, b_init, c_init, v_init = initial_vertex_polygon_dimensions(A, b, I)
B_init = next(iter(bases(v_init, A_init, b_init)))
_, v_start1, _, _ = simplex(A_init, b_init, c_init, v_init, B_init, pivot_rule_i=PivotRule.MINIMAL())
v_start1 = Matrix(v_start1[:2])
I_start1 = active_constraints(v_start1, A, b)
assert v_start1 == Matrix([1,0])
assert set(I_start1) == {2, 3}
def test_ex121():
c = Matrix([-1, -1, 1])
b = Matrix([6+Rational(4,3), 4+Rational(2,3), 6, 4, 0, 0, 0])
A = Matrix([
[1, 2, 0],
[1, 1, 1],
[3, 0, 1],
[0, 0, 1],
[-1, 0, 0],
[0, -1, 0],
[0, 0, -1]
])
x_bar_0 = Matrix([2, 2+Rational(2,3), 0])
assert is_contained(x_bar_0, A, b)
assert set(active_constraints(x_bar_0, A, b)) == {0, 1, 2, 6}
B = list(bases(x_bar_0,A,b))[-1]
res, x_star, opt_val, _ = simplex(A, b, c, x_bar_0, set(B), pivot_rule_i=PivotRule.MAXIMAL(), pivot_rule_p=PivotRule.MAXIMAL())
assert x_star == Matrix([0, 0, 4])
def test_cycle():
# more examples: http://web.ist.utl.pt/~mcasquilho/CD_Casquilho/LP2004Comp&OR_GassVinjamuri.pdf
A = Matrix([
[-1, 0, 0],
[0, -1, 0],
[1, 1, -1],
[-4, -1, -2],
[1, -3, -3],
[3, 4, -6],
[0, 0, 1]
])
b = Matrix(6*[0]+[1])
c = Matrix([0, 0, 1])
def custom_pivot(xs: List[int], *args, **kwargs):
if xs == [2,5]:
return 5
if xs == [0,3]:
return 3
if xs == [1,4]:
return 4
return xs[0]
res, _, _, _ = simplex(A, b, c, Matrix([0,0,0]), {0,1,4}, pivot_rule_i=custom_pivot)
assert res == SimplexResult.CYCLE
def test_example3428():
A = Matrix([
[1, 2, 1],
[-2, 1, 0],
[1, 0, 0],
[0, 1, 0],
[0, 0, 1],
[-1, 0, 0],
[0, -1, 0],
[0, 0, -1]
])
m, n = A.shape
b = Matrix([3, 0, 1, 1, 1, 0, 0, 0])
c = Matrix([1,1,1])
B = {2,6,7}
v_3 = Matrix([1,0,0])
simplex(A, b, c, v_3, B, pivot_rule_i=PivotRule.MINIMAL())
# two perturbations are applied
r1 = range(m)
r2 = [5, 4, 3, 1, 0, 7, 6, 2]
# once with explicit pertubation
e = pertubation_vector(r1, Rational(1,64))
v_3_e = v_3 + (sub_matrix(A, B)**-1*sub_matrix(e, B))
b_e = b + e
res, v_r1_star_expl_pert, _, _ = simplex(A, b_e, c, v_3_e, B, pivot_rule_i=PivotRule.MINIMAL())
v_r1_star_expl = v_star_from_perturbed_polygon(A, b, b_e, v_r1_star_expl_pert)
e = pertubation_vector(r2, Rational(1,64))
v_3_e = v_3 + (sub_matrix(A, B)**-1*sub_matrix(e, B))
b_e = b + e
res, v_r2_star_expl_pert, _, _= simplex(A, b_e, c, v_3_e, B, pivot_rule_i=PivotRule.MINIMAL())
v_r2_star_expl = v_star_from_perturbed_polygon(A, b, b_e, v_r2_star_expl_pert)
# and once with our fancy lexmin rule
res, v_r1_star_lexmin, _, _ = simplex(A, b, c, v_3, B, pivot_rule_i=PivotRule.LEXMIN(r1))
res, v_r2_star_lexmin, _, _ = simplex(A, b, c, v_3, B, pivot_rule_i=PivotRule.LEXMIN(r2))
assert v_r1_star_expl == v_r1_star_lexmin
assert v_r2_star_expl == v_r2_star_lexmin