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AIM-PIbd-32-Kurbanova-A-A/aimenv/Lib/site-packages/scipy/optimize/tests/test_linprog.py

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lab 1 is done
2024-10-02 22:15:59 +04:00
"""
Unit test for Linear Programming
"""
import sys
import platform
import numpy as np
from numpy.testing import (assert_, assert_allclose, assert_equal,
assert_array_less, assert_warns, suppress_warnings)
from pytest import raises as assert_raises
from scipy.optimize import linprog, OptimizeWarning
from scipy.optimize._numdiff import approx_derivative
from scipy.sparse.linalg import MatrixRankWarning
from scipy.linalg import LinAlgWarning
from scipy._lib._util import VisibleDeprecationWarning
import scipy.sparse
import pytest
has_umfpack = True
try:
from scikits.umfpack import UmfpackWarning
except ImportError:
has_umfpack = False
has_cholmod = True
try:
import sksparse # noqa: F401
from sksparse.cholmod import cholesky as cholmod # noqa: F401
except ImportError:
has_cholmod = False
def _assert_iteration_limit_reached(res, maxiter):
assert_(not res.success, "Incorrectly reported success")
assert_(res.success < maxiter, "Incorrectly reported number of iterations")
assert_equal(res.status, 1, "Failed to report iteration limit reached")
def _assert_infeasible(res):
# res: linprog result object
assert_(not res.success, "incorrectly reported success")
assert_equal(res.status, 2, "failed to report infeasible status")
def _assert_unbounded(res):
# res: linprog result object
assert_(not res.success, "incorrectly reported success")
assert_equal(res.status, 3, "failed to report unbounded status")
def _assert_unable_to_find_basic_feasible_sol(res):
# res: linprog result object
# The status may be either 2 or 4 depending on why the feasible solution
# could not be found. If the underlying problem is expected to not have a
# feasible solution, _assert_infeasible should be used.
assert_(not res.success, "incorrectly reported success")
assert_(res.status in (2, 4), "failed to report optimization failure")
def _assert_success(res, desired_fun=None, desired_x=None,
rtol=1e-8, atol=1e-8):
# res: linprog result object
# desired_fun: desired objective function value or None
# desired_x: desired solution or None
if not res.success:
msg = f"linprog status {res.status}, message: {res.message}"
raise AssertionError(msg)
assert_equal(res.status, 0)
if desired_fun is not None:
assert_allclose(res.fun, desired_fun,
err_msg="converged to an unexpected objective value",
rtol=rtol, atol=atol)
if desired_x is not None:
assert_allclose(res.x, desired_x,
err_msg="converged to an unexpected solution",
rtol=rtol, atol=atol)
def magic_square(n):
"""
Generates a linear program for which integer solutions represent an
n x n magic square; binary decision variables represent the presence
(or absence) of an integer 1 to n^2 in each position of the square.
"""
np.random.seed(0)
M = n * (n**2 + 1) / 2
numbers = np.arange(n**4) // n**2 + 1
numbers = numbers.reshape(n**2, n, n)
zeros = np.zeros((n**2, n, n))
A_list = []
b_list = []
# Rule 1: use every number exactly once
for i in range(n**2):
A_row = zeros.copy()
A_row[i, :, :] = 1
A_list.append(A_row.flatten())
b_list.append(1)
# Rule 2: Only one number per square
for i in range(n):
for j in range(n):
A_row = zeros.copy()
A_row[:, i, j] = 1
A_list.append(A_row.flatten())
b_list.append(1)
# Rule 3: sum of rows is M
for i in range(n):
A_row = zeros.copy()
A_row[:, i, :] = numbers[:, i, :]
A_list.append(A_row.flatten())
b_list.append(M)
# Rule 4: sum of columns is M
for i in range(n):
A_row = zeros.copy()
A_row[:, :, i] = numbers[:, :, i]
A_list.append(A_row.flatten())
b_list.append(M)
# Rule 5: sum of diagonals is M
A_row = zeros.copy()
A_row[:, range(n), range(n)] = numbers[:, range(n), range(n)]
A_list.append(A_row.flatten())
b_list.append(M)
A_row = zeros.copy()
A_row[:, range(n), range(-1, -n - 1, -1)] = \
numbers[:, range(n), range(-1, -n - 1, -1)]
A_list.append(A_row.flatten())
b_list.append(M)
A = np.array(np.vstack(A_list), dtype=float)
b = np.array(b_list, dtype=float)
c = np.random.rand(A.shape[1])
return A, b, c, numbers, M
def lpgen_2d(m, n):
""" -> A b c LP test: m*n vars, m+n constraints
row sums == n/m, col sums == 1
https://gist.github.com/denis-bz/8647461
"""
np.random.seed(0)
c = - np.random.exponential(size=(m, n))
Arow = np.zeros((m, m * n))
brow = np.zeros(m)
for j in range(m):
j1 = j + 1
Arow[j, j * n:j1 * n] = 1
brow[j] = n / m
Acol = np.zeros((n, m * n))
bcol = np.zeros(n)
for j in range(n):
j1 = j + 1
Acol[j, j::n] = 1
bcol[j] = 1
A = np.vstack((Arow, Acol))
b = np.hstack((brow, bcol))
return A, b, c.ravel()
def very_random_gen(seed=0):
np.random.seed(seed)
m_eq, m_ub, n = 10, 20, 50
c = np.random.rand(n)-0.5
A_ub = np.random.rand(m_ub, n)-0.5
b_ub = np.random.rand(m_ub)-0.5
A_eq = np.random.rand(m_eq, n)-0.5
b_eq = np.random.rand(m_eq)-0.5
lb = -np.random.rand(n)
ub = np.random.rand(n)
lb[lb < -np.random.rand()] = -np.inf
ub[ub > np.random.rand()] = np.inf
bounds = np.vstack((lb, ub)).T
return c, A_ub, b_ub, A_eq, b_eq, bounds
def nontrivial_problem():
c = [-1, 8, 4, -6]
A_ub = [[-7, -7, 6, 9],
[1, -1, -3, 0],
[10, -10, -7, 7],
[6, -1, 3, 4]]
b_ub = [-3, 6, -6, 6]
A_eq = [[-10, 1, 1, -8]]
b_eq = [-4]
x_star = [101 / 1391, 1462 / 1391, 0, 752 / 1391]
f_star = 7083 / 1391
return c, A_ub, b_ub, A_eq, b_eq, x_star, f_star
def l1_regression_prob(seed=0, m=8, d=9, n=100):
'''
Training data is {(x0, y0), (x1, y2), ..., (xn-1, yn-1)}
x in R^d
y in R
n: number of training samples
d: dimension of x, i.e. x in R^d
phi: feature map R^d -> R^m
m: dimension of feature space
'''
np.random.seed(seed)
phi = np.random.normal(0, 1, size=(m, d)) # random feature mapping
w_true = np.random.randn(m)
x = np.random.normal(0, 1, size=(d, n)) # features
y = w_true @ (phi @ x) + np.random.normal(0, 1e-5, size=n) # measurements
# construct the problem
c = np.ones(m+n)
c[:m] = 0
A_ub = scipy.sparse.lil_matrix((2*n, n+m))
idx = 0
for ii in range(n):
A_ub[idx, :m] = phi @ x[:, ii]
A_ub[idx, m+ii] = -1
A_ub[idx+1, :m] = -1*phi @ x[:, ii]
A_ub[idx+1, m+ii] = -1
idx += 2
A_ub = A_ub.tocsc()
b_ub = np.zeros(2*n)
b_ub[0::2] = y
b_ub[1::2] = -y
bnds = [(None, None)]*m + [(0, None)]*n
return c, A_ub, b_ub, bnds
def generic_callback_test(self):
# Check that callback is as advertised
last_cb = {}
def cb(res):
message = res.pop('message')
complete = res.pop('complete')
assert_(res.pop('phase') in (1, 2))
assert_(res.pop('status') in range(4))
assert_(isinstance(res.pop('nit'), int))
assert_(isinstance(complete, bool))
assert_(isinstance(message, str))
last_cb['x'] = res['x']
last_cb['fun'] = res['fun']
last_cb['slack'] = res['slack']
last_cb['con'] = res['con']
c = np.array([-3, -2])
A_ub = [[2, 1], [1, 1], [1, 0]]
b_ub = [10, 8, 4]
res = linprog(c, A_ub=A_ub, b_ub=b_ub, callback=cb, method=self.method)
_assert_success(res, desired_fun=-18.0, desired_x=[2, 6])
assert_allclose(last_cb['fun'], res['fun'])
assert_allclose(last_cb['x'], res['x'])
assert_allclose(last_cb['con'], res['con'])
assert_allclose(last_cb['slack'], res['slack'])
def test_unknown_solvers_and_options():
c = np.array([-3, -2])
A_ub = [[2, 1], [1, 1], [1, 0]]
b_ub = [10, 8, 4]
assert_raises(ValueError, linprog,
c, A_ub=A_ub, b_ub=b_ub, method='ekki-ekki-ekki')
assert_raises(ValueError, linprog,
c, A_ub=A_ub, b_ub=b_ub, method='highs-ekki')
message = "Unrecognized options detected: {'rr_method': 'ekki-ekki-ekki'}"
with pytest.warns(OptimizeWarning, match=message):
linprog(c, A_ub=A_ub, b_ub=b_ub,
options={"rr_method": 'ekki-ekki-ekki'})
def test_choose_solver():
# 'highs' chooses 'dual'
c = np.array([-3, -2])
A_ub = [[2, 1], [1, 1], [1, 0]]
b_ub = [10, 8, 4]
res = linprog(c, A_ub, b_ub, method='highs')
_assert_success(res, desired_fun=-18.0, desired_x=[2, 6])
def test_deprecation():
with pytest.warns(DeprecationWarning):
linprog(1, method='interior-point')
with pytest.warns(DeprecationWarning):
linprog(1, method='revised simplex')
with pytest.warns(DeprecationWarning):
linprog(1, method='simplex')
def test_highs_status_message():
res = linprog(1, method='highs')
msg = "Optimization terminated successfully. (HiGHS Status 7:"
assert res.status == 0
assert res.message.startswith(msg)
A, b, c, numbers, M = magic_square(6)
bounds = [(0, 1)] * len(c)
integrality = [1] * len(c)
options = {"time_limit": 0.1}
res = linprog(c=c, A_eq=A, b_eq=b, bounds=bounds, method='highs',
options=options, integrality=integrality)
msg = "Time limit reached. (HiGHS Status 13:"
assert res.status == 1
assert res.message.startswith(msg)
options = {"maxiter": 10}
res = linprog(c=c, A_eq=A, b_eq=b, bounds=bounds, method='highs-ds',
options=options)
msg = "Iteration limit reached. (HiGHS Status 14:"
assert res.status == 1
assert res.message.startswith(msg)
res = linprog(1, bounds=(1, -1), method='highs')
msg = "The problem is infeasible. (HiGHS Status 8:"
assert res.status == 2
assert res.message.startswith(msg)
res = linprog(-1, method='highs')
msg = "The problem is unbounded. (HiGHS Status 10:"
assert res.status == 3
assert res.message.startswith(msg)
from scipy.optimize._linprog_highs import _highs_to_scipy_status_message
status, message = _highs_to_scipy_status_message(58, "Hello!")
msg = "The HiGHS status code was not recognized. (HiGHS Status 58:"
assert status == 4
assert message.startswith(msg)
status, message = _highs_to_scipy_status_message(None, None)
msg = "HiGHS did not provide a status code. (HiGHS Status None: None)"
assert status == 4
assert message.startswith(msg)
def test_bug_17380():
linprog([1, 1], A_ub=[[-1, 0]], b_ub=[-2.5], integrality=[1, 1])
A_ub = None
b_ub = None
A_eq = None
b_eq = None
bounds = None
################
# Common Tests #
################
class LinprogCommonTests:
"""
Base class for `linprog` tests. Generally, each test will be performed
once for every derived class of LinprogCommonTests, each of which will
typically change self.options and/or self.method. Effectively, these tests
are run for many combination of method (simplex, revised simplex, and
interior point) and options (such as pivoting rule or sparse treatment).
"""
##################
# Targeted Tests #
##################
def test_callback(self):
generic_callback_test(self)
def test_disp(self):
# test that display option does not break anything.
A, b, c = lpgen_2d(20, 20)
res = linprog(c, A_ub=A, b_ub=b, method=self.method,
options={"disp": True})
_assert_success(res, desired_fun=-64.049494229)
def test_docstring_example(self):
# Example from linprog docstring.
c = [-1, 4]
A = [[-3, 1], [1, 2]]
b = [6, 4]
x0_bounds = (None, None)
x1_bounds = (-3, None)
res = linprog(c, A_ub=A, b_ub=b, bounds=(x0_bounds, x1_bounds),
options=self.options, method=self.method)
_assert_success(res, desired_fun=-22)
def test_type_error(self):
# (presumably) checks that linprog recognizes type errors
# This is tested more carefully in test__linprog_clean_inputs.py
c = [1]
A_eq = [[1]]
b_eq = "hello"
assert_raises(TypeError, linprog,
c, A_eq=A_eq, b_eq=b_eq,
method=self.method, options=self.options)
def test_aliasing_b_ub(self):
# (presumably) checks that linprog does not modify b_ub
# This is tested more carefully in test__linprog_clean_inputs.py
c = np.array([1.0])
A_ub = np.array([[1.0]])
b_ub_orig = np.array([3.0])
b_ub = b_ub_orig.copy()
bounds = (-4.0, np.inf)
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_fun=-4, desired_x=[-4])
assert_allclose(b_ub_orig, b_ub)
def test_aliasing_b_eq(self):
# (presumably) checks that linprog does not modify b_eq
# This is tested more carefully in test__linprog_clean_inputs.py
c = np.array([1.0])
A_eq = np.array([[1.0]])
b_eq_orig = np.array([3.0])
b_eq = b_eq_orig.copy()
bounds = (-4.0, np.inf)
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_fun=3, desired_x=[3])
assert_allclose(b_eq_orig, b_eq)
def test_non_ndarray_args(self):
# (presumably) checks that linprog accepts list in place of arrays
# This is tested more carefully in test__linprog_clean_inputs.py
c = [1.0]
A_ub = [[1.0]]
b_ub = [3.0]
A_eq = [[1.0]]
b_eq = [2.0]
bounds = (-1.0, 10.0)
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_fun=2, desired_x=[2])
def test_unknown_options(self):
c = np.array([-3, -2])
A_ub = [[2, 1], [1, 1], [1, 0]]
b_ub = [10, 8, 4]
def f(c, A_ub=None, b_ub=None, A_eq=None,
b_eq=None, bounds=None, options={}):
linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=options)
o = {key: self.options[key] for key in self.options}
o['spam'] = 42
assert_warns(OptimizeWarning, f,
c, A_ub=A_ub, b_ub=b_ub, options=o)
def test_integrality_without_highs(self):
# ensure that using `integrality` parameter without `method='highs'`
# raises warning and produces correct solution to relaxed problem
# source: https://en.wikipedia.org/wiki/Integer_programming#Example
A_ub = np.array([[-1, 1], [3, 2], [2, 3]])
b_ub = np.array([1, 12, 12])
c = -np.array([0, 1])
bounds = [(0, np.inf)] * len(c)
integrality = [1] * len(c)
with np.testing.assert_warns(OptimizeWarning):
res = linprog(c=c, A_ub=A_ub, b_ub=b_ub, bounds=bounds,
method=self.method, integrality=integrality)
np.testing.assert_allclose(res.x, [1.8, 2.8])
np.testing.assert_allclose(res.fun, -2.8)
def test_invalid_inputs(self):
def f(c, A_ub=None, b_ub=None, A_eq=None, b_eq=None, bounds=None):
linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
# Test ill-formatted bounds
assert_raises(ValueError, f, [1, 2, 3], bounds=[(1, 2), (3, 4)])
with np.testing.suppress_warnings() as sup:
sup.filter(VisibleDeprecationWarning, "Creating an ndarray from ragged")
assert_raises(ValueError, f, [1, 2, 3], bounds=[(1, 2), (3, 4), (3, 4, 5)])
assert_raises(ValueError, f, [1, 2, 3], bounds=[(1, -2), (1, 2)])
# Test other invalid inputs
assert_raises(ValueError, f, [1, 2], A_ub=[[1, 2]], b_ub=[1, 2])
assert_raises(ValueError, f, [1, 2], A_ub=[[1]], b_ub=[1])
assert_raises(ValueError, f, [1, 2], A_eq=[[1, 2]], b_eq=[1, 2])
assert_raises(ValueError, f, [1, 2], A_eq=[[1]], b_eq=[1])
assert_raises(ValueError, f, [1, 2], A_eq=[1], b_eq=1)
# this last check doesn't make sense for sparse presolve
if ("_sparse_presolve" in self.options and
self.options["_sparse_presolve"]):
return
# there aren't 3-D sparse matrices
assert_raises(ValueError, f, [1, 2], A_ub=np.zeros((1, 1, 3)), b_eq=1)
def test_sparse_constraints(self):
# gh-13559: improve error message for sparse inputs when unsupported
def f(c, A_ub=None, b_ub=None, A_eq=None, b_eq=None, bounds=None):
linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
np.random.seed(0)
m = 100
n = 150
A_eq = scipy.sparse.rand(m, n, 0.5)
x_valid = np.random.randn(n)
c = np.random.randn(n)
ub = x_valid + np.random.rand(n)
lb = x_valid - np.random.rand(n)
bounds = np.column_stack((lb, ub))
b_eq = A_eq * x_valid
if self.method in {'simplex', 'revised simplex'}:
# simplex and revised simplex should raise error
with assert_raises(ValueError, match=f"Method '{self.method}' "
"does not support sparse constraint matrices."):
linprog(c=c, A_eq=A_eq, b_eq=b_eq, bounds=bounds,
method=self.method, options=self.options)
else:
# other methods should succeed
options = {**self.options}
if self.method in {'interior-point'}:
options['sparse'] = True
res = linprog(c=c, A_eq=A_eq, b_eq=b_eq, bounds=bounds,
method=self.method, options=options)
assert res.success
def test_maxiter(self):
# test iteration limit w/ Enzo example
c = [4, 8, 3, 0, 0, 0]
A = [
[2, 5, 3, -1, 0, 0],
[3, 2.5, 8, 0, -1, 0],
[8, 10, 4, 0, 0, -1]]
b = [185, 155, 600]
np.random.seed(0)
maxiter = 3
res = linprog(c, A_eq=A, b_eq=b, method=self.method,
options={"maxiter": maxiter})
_assert_iteration_limit_reached(res, maxiter)
assert_equal(res.nit, maxiter)
def test_bounds_fixed(self):
# Test fixed bounds (upper equal to lower)
# If presolve option True, test if solution found in presolve (i.e.
# number of iterations is 0).
do_presolve = self.options.get('presolve', True)
res = linprog([1], bounds=(1, 1),
method=self.method, options=self.options)
_assert_success(res, 1, 1)
if do_presolve:
assert_equal(res.nit, 0)
res = linprog([1, 2, 3], bounds=[(5, 5), (-1, -1), (3, 3)],
method=self.method, options=self.options)
_assert_success(res, 12, [5, -1, 3])
if do_presolve:
assert_equal(res.nit, 0)
res = linprog([1, 1], bounds=[(1, 1), (1, 3)],
method=self.method, options=self.options)
_assert_success(res, 2, [1, 1])
if do_presolve:
assert_equal(res.nit, 0)
res = linprog([1, 1, 2], A_eq=[[1, 0, 0], [0, 1, 0]], b_eq=[1, 7],
bounds=[(-5, 5), (0, 10), (3.5, 3.5)],
method=self.method, options=self.options)
_assert_success(res, 15, [1, 7, 3.5])
if do_presolve:
assert_equal(res.nit, 0)
def test_bounds_infeasible(self):
# Test ill-valued bounds (upper less than lower)
# If presolve option True, test if solution found in presolve (i.e.
# number of iterations is 0).
do_presolve = self.options.get('presolve', True)
res = linprog([1], bounds=(1, -2), method=self.method, options=self.options)
_assert_infeasible(res)
if do_presolve:
assert_equal(res.nit, 0)
res = linprog([1], bounds=[(1, -2)], method=self.method, options=self.options)
_assert_infeasible(res)
if do_presolve:
assert_equal(res.nit, 0)
res = linprog([1, 2, 3], bounds=[(5, 0), (1, 2), (3, 4)],
method=self.method, options=self.options)
_assert_infeasible(res)
if do_presolve:
assert_equal(res.nit, 0)
def test_bounds_infeasible_2(self):
# Test ill-valued bounds (lower inf, upper -inf)
# If presolve option True, test if solution found in presolve (i.e.
# number of iterations is 0).
# For the simplex method, the cases do not result in an
# infeasible status, but in a RuntimeWarning. This is a
# consequence of having _presolve() take care of feasibility
# checks. See issue gh-11618.
do_presolve = self.options.get('presolve', True)
simplex_without_presolve = not do_presolve and self.method == 'simplex'
c = [1, 2, 3]
bounds_1 = [(1, 2), (np.inf, np.inf), (3, 4)]
bounds_2 = [(1, 2), (-np.inf, -np.inf), (3, 4)]
if simplex_without_presolve:
def g(c, bounds):
res = linprog(c, bounds=bounds,
method=self.method, options=self.options)
return res
with pytest.warns(RuntimeWarning):
with pytest.raises(IndexError):
g(c, bounds=bounds_1)
with pytest.warns(RuntimeWarning):
with pytest.raises(IndexError):
g(c, bounds=bounds_2)
else:
res = linprog(c=c, bounds=bounds_1,
method=self.method, options=self.options)
_assert_infeasible(res)
if do_presolve:
assert_equal(res.nit, 0)
res = linprog(c=c, bounds=bounds_2,
method=self.method, options=self.options)
_assert_infeasible(res)
if do_presolve:
assert_equal(res.nit, 0)
def test_empty_constraint_1(self):
c = [-1, -2]
res = linprog(c, method=self.method, options=self.options)
_assert_unbounded(res)
def test_empty_constraint_2(self):
c = [-1, 1, -1, 1]
bounds = [(0, np.inf), (-np.inf, 0), (-1, 1), (-1, 1)]
res = linprog(c, bounds=bounds,
method=self.method, options=self.options)
_assert_unbounded(res)
# Unboundedness detected in presolve requires no iterations
if self.options.get('presolve', True):
assert_equal(res.nit, 0)
def test_empty_constraint_3(self):
c = [1, -1, 1, -1]
bounds = [(0, np.inf), (-np.inf, 0), (-1, 1), (-1, 1)]
res = linprog(c, bounds=bounds,
method=self.method, options=self.options)
_assert_success(res, desired_x=[0, 0, -1, 1], desired_fun=-2)
def test_inequality_constraints(self):
# Minimize linear function subject to linear inequality constraints.
# http://www.dam.brown.edu/people/huiwang/classes/am121/Archive/simplex_121_c.pdf
c = np.array([3, 2]) * -1 # maximize
A_ub = [[2, 1],
[1, 1],
[1, 0]]
b_ub = [10, 8, 4]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_fun=-18, desired_x=[2, 6])
def test_inequality_constraints2(self):
# Minimize linear function subject to linear inequality constraints.
# http://www.statslab.cam.ac.uk/~ff271/teaching/opt/notes/notes8.pdf
# (dead link)
c = [6, 3]
A_ub = [[0, 3],
[-1, -1],
[-2, 1]]
b_ub = [2, -1, -1]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_fun=5, desired_x=[2 / 3, 1 / 3])
def test_bounds_simple(self):
c = [1, 2]
bounds = (1, 2)
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_x=[1, 1])
bounds = [(1, 2), (1, 2)]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_x=[1, 1])
def test_bounded_below_only_1(self):
c = np.array([1.0])
A_eq = np.array([[1.0]])
b_eq = np.array([3.0])
bounds = (1.0, None)
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_fun=3, desired_x=[3])
def test_bounded_below_only_2(self):
c = np.ones(3)
A_eq = np.eye(3)
b_eq = np.array([1, 2, 3])
bounds = (0.5, np.inf)
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_x=b_eq, desired_fun=np.sum(b_eq))
def test_bounded_above_only_1(self):
c = np.array([1.0])
A_eq = np.array([[1.0]])
b_eq = np.array([3.0])
bounds = (None, 10.0)
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_fun=3, desired_x=[3])
def test_bounded_above_only_2(self):
c = np.ones(3)
A_eq = np.eye(3)
b_eq = np.array([1, 2, 3])
bounds = (-np.inf, 4)
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_x=b_eq, desired_fun=np.sum(b_eq))
def test_bounds_infinity(self):
c = np.ones(3)
A_eq = np.eye(3)
b_eq = np.array([1, 2, 3])
bounds = (-np.inf, np.inf)
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_x=b_eq, desired_fun=np.sum(b_eq))
def test_bounds_mixed(self):
# Problem has one unbounded variable and
# another with a negative lower bound.
c = np.array([-1, 4]) * -1 # maximize
A_ub = np.array([[-3, 1],
[1, 2]], dtype=np.float64)
b_ub = [6, 4]
x0_bounds = (-np.inf, np.inf)
x1_bounds = (-3, np.inf)
bounds = (x0_bounds, x1_bounds)
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_fun=-80 / 7, desired_x=[-8 / 7, 18 / 7])
def test_bounds_equal_but_infeasible(self):
c = [-4, 1]
A_ub = [[7, -2], [0, 1], [2, -2]]
b_ub = [14, 0, 3]
bounds = [(2, 2), (0, None)]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_infeasible(res)
def test_bounds_equal_but_infeasible2(self):
c = [-4, 1]
A_eq = [[7, -2], [0, 1], [2, -2]]
b_eq = [14, 0, 3]
bounds = [(2, 2), (0, None)]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_infeasible(res)
def test_bounds_equal_no_presolve(self):
# There was a bug when a lower and upper bound were equal but
# presolve was not on to eliminate the variable. The bound
# was being converted to an equality constraint, but the bound
# was not eliminated, leading to issues in postprocessing.
c = [1, 2]
A_ub = [[1, 2], [1.1, 2.2]]
b_ub = [4, 8]
bounds = [(1, 2), (2, 2)]
o = {key: self.options[key] for key in self.options}
o["presolve"] = False
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=o)
_assert_infeasible(res)
def test_zero_column_1(self):
m, n = 3, 4
np.random.seed(0)
c = np.random.rand(n)
c[1] = 1
A_eq = np.random.rand(m, n)
A_eq[:, 1] = 0
b_eq = np.random.rand(m)
A_ub = [[1, 0, 1, 1]]
b_ub = 3
bounds = [(-10, 10), (-10, 10), (-10, None), (None, None)]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_fun=-9.7087836730413404)
def test_zero_column_2(self):
if self.method in {'highs-ds', 'highs-ipm'}:
# See upstream issue https://github.com/ERGO-Code/HiGHS/issues/648
pytest.xfail()
np.random.seed(0)
m, n = 2, 4
c = np.random.rand(n)
c[1] = -1
A_eq = np.random.rand(m, n)
A_eq[:, 1] = 0
b_eq = np.random.rand(m)
A_ub = np.random.rand(m, n)
A_ub[:, 1] = 0
b_ub = np.random.rand(m)
bounds = (None, None)
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_unbounded(res)
# Unboundedness detected in presolve
if self.options.get('presolve', True) and "highs" not in self.method:
# HiGHS detects unboundedness or infeasibility in presolve
# It needs an iteration of simplex to be sure of unboundedness
# Other solvers report that the problem is unbounded if feasible
assert_equal(res.nit, 0)
def test_zero_row_1(self):
c = [1, 2, 3]
A_eq = [[0, 0, 0], [1, 1, 1], [0, 0, 0]]
b_eq = [0, 3, 0]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_fun=3)
def test_zero_row_2(self):
A_ub = [[0, 0, 0], [1, 1, 1], [0, 0, 0]]
b_ub = [0, 3, 0]
c = [1, 2, 3]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_fun=0)
def test_zero_row_3(self):
m, n = 2, 4
c = np.random.rand(n)
A_eq = np.random.rand(m, n)
A_eq[0, :] = 0
b_eq = np.random.rand(m)
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_infeasible(res)
# Infeasibility detected in presolve
if self.options.get('presolve', True):
assert_equal(res.nit, 0)
def test_zero_row_4(self):
m, n = 2, 4
c = np.random.rand(n)
A_ub = np.random.rand(m, n)
A_ub[0, :] = 0
b_ub = -np.random.rand(m)
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_infeasible(res)
# Infeasibility detected in presolve
if self.options.get('presolve', True):
assert_equal(res.nit, 0)
def test_singleton_row_eq_1(self):
c = [1, 1, 1, 2]
A_eq = [[1, 0, 0, 0], [0, 2, 0, 0], [1, 0, 0, 0], [1, 1, 1, 1]]
b_eq = [1, 2, 2, 4]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_infeasible(res)
# Infeasibility detected in presolve
if self.options.get('presolve', True):
assert_equal(res.nit, 0)
def test_singleton_row_eq_2(self):
c = [1, 1, 1, 2]
A_eq = [[1, 0, 0, 0], [0, 2, 0, 0], [1, 0, 0, 0], [1, 1, 1, 1]]
b_eq = [1, 2, 1, 4]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_fun=4)
def test_singleton_row_ub_1(self):
c = [1, 1, 1, 2]
A_ub = [[1, 0, 0, 0], [0, 2, 0, 0], [-1, 0, 0, 0], [1, 1, 1, 1]]
b_ub = [1, 2, -2, 4]
bounds = [(None, None), (0, None), (0, None), (0, None)]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_infeasible(res)
# Infeasibility detected in presolve
if self.options.get('presolve', True):
assert_equal(res.nit, 0)
def test_singleton_row_ub_2(self):
c = [1, 1, 1, 2]
A_ub = [[1, 0, 0, 0], [0, 2, 0, 0], [-1, 0, 0, 0], [1, 1, 1, 1]]
b_ub = [1, 2, -0.5, 4]
bounds = [(None, None), (0, None), (0, None), (0, None)]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_fun=0.5)
def test_infeasible(self):
# Test linprog response to an infeasible problem
c = [-1, -1]
A_ub = [[1, 0],
[0, 1],
[-1, -1]]
b_ub = [2, 2, -5]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_infeasible(res)
def test_infeasible_inequality_bounds(self):
c = [1]
A_ub = [[2]]
b_ub = 4
bounds = (5, 6)
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_infeasible(res)
# Infeasibility detected in presolve
if self.options.get('presolve', True):
assert_equal(res.nit, 0)
def test_unbounded(self):
# Test linprog response to an unbounded problem
c = np.array([1, 1]) * -1 # maximize
A_ub = [[-1, 1],
[-1, -1]]
b_ub = [-1, -2]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_unbounded(res)
def test_unbounded_below_no_presolve_corrected(self):
c = [1]
bounds = [(None, 1)]
o = {key: self.options[key] for key in self.options}
o["presolve"] = False
res = linprog(c=c, bounds=bounds,
method=self.method,
options=o)
if self.method == "revised simplex":
# Revised simplex has a special pathway for no constraints.
assert_equal(res.status, 5)
else:
_assert_unbounded(res)
def test_unbounded_no_nontrivial_constraints_1(self):
"""
Test whether presolve pathway for detecting unboundedness after
constraint elimination is working.
"""
c = np.array([0, 0, 0, 1, -1, -1])
A_ub = np.array([[1, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0],
[0, 0, 0, 0, 0, -1]])
b_ub = np.array([2, -2, 0])
bounds = [(None, None), (None, None), (None, None),
(-1, 1), (-1, 1), (0, None)]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_unbounded(res)
if not self.method.lower().startswith("highs"):
assert_equal(res.x[-1], np.inf)
assert_equal(res.message[:36],
"The problem is (trivially) unbounded")
def test_unbounded_no_nontrivial_constraints_2(self):
"""
Test whether presolve pathway for detecting unboundedness after
constraint elimination is working.
"""
c = np.array([0, 0, 0, 1, -1, 1])
A_ub = np.array([[1, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 1]])
b_ub = np.array([2, -2, 0])
bounds = [(None, None), (None, None), (None, None),
(-1, 1), (-1, 1), (None, 0)]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_unbounded(res)
if not self.method.lower().startswith("highs"):
assert_equal(res.x[-1], -np.inf)
assert_equal(res.message[:36],
"The problem is (trivially) unbounded")
def test_cyclic_recovery(self):
# Test linprogs recovery from cycling using the Klee-Minty problem
# Klee-Minty https://www.math.ubc.ca/~israel/m340/kleemin3.pdf
c = np.array([100, 10, 1]) * -1 # maximize
A_ub = [[1, 0, 0],
[20, 1, 0],
[200, 20, 1]]
b_ub = [1, 100, 10000]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_x=[0, 0, 10000], atol=5e-6, rtol=1e-7)
def test_cyclic_bland(self):
# Test the effect of Bland's rule on a cycling problem
c = np.array([-10, 57, 9, 24.])
A_ub = np.array([[0.5, -5.5, -2.5, 9],
[0.5, -1.5, -0.5, 1],
[1, 0, 0, 0]])
b_ub = [0, 0, 1]
# copy the existing options dictionary but change maxiter
maxiter = 100
o = {key: val for key, val in self.options.items()}
o['maxiter'] = maxiter
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=o)
if self.method == 'simplex' and not self.options.get('bland'):
# simplex cycles without Bland's rule
_assert_iteration_limit_reached(res, o['maxiter'])
else:
# other methods, including simplex with Bland's rule, succeed
_assert_success(res, desired_x=[1, 0, 1, 0])
# note that revised simplex skips this test because it may or may not
# cycle depending on the initial basis
def test_remove_redundancy_infeasibility(self):
# mostly a test of redundancy removal, which is carefully tested in
# test__remove_redundancy.py
m, n = 10, 10
c = np.random.rand(n)
A_eq = np.random.rand(m, n)
b_eq = np.random.rand(m)
A_eq[-1, :] = 2 * A_eq[-2, :]
b_eq[-1] *= -1
with suppress_warnings() as sup:
sup.filter(OptimizeWarning, "A_eq does not appear...")
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_infeasible(res)
#################
# General Tests #
#################
def test_nontrivial_problem(self):
# Problem involves all constraint types,
# negative resource limits, and rounding issues.
c, A_ub, b_ub, A_eq, b_eq, x_star, f_star = nontrivial_problem()
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_fun=f_star, desired_x=x_star)
def test_lpgen_problem(self):
# Test linprog with a rather large problem (400 variables,
# 40 constraints) generated by https://gist.github.com/denis-bz/8647461
A_ub, b_ub, c = lpgen_2d(20, 20)
with suppress_warnings() as sup:
sup.filter(OptimizeWarning, "Solving system with option 'sym_pos'")
sup.filter(RuntimeWarning, "invalid value encountered")
sup.filter(LinAlgWarning)
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_fun=-64.049494229)
def test_network_flow(self):
# A network flow problem with supply and demand at nodes
# and with costs along directed edges.
# https://www.princeton.edu/~rvdb/542/lectures/lec10.pdf
c = [2, 4, 9, 11, 4, 3, 8, 7, 0, 15, 16, 18]
n, p = -1, 1
A_eq = [
[n, n, p, 0, p, 0, 0, 0, 0, p, 0, 0],
[p, 0, 0, p, 0, p, 0, 0, 0, 0, 0, 0],
[0, 0, n, n, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, p, p, 0, 0, p, 0],
[0, 0, 0, 0, n, n, n, 0, p, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, n, n, 0, 0, p],
[0, 0, 0, 0, 0, 0, 0, 0, 0, n, n, n]]
b_eq = [0, 19, -16, 33, 0, 0, -36]
with suppress_warnings() as sup:
sup.filter(LinAlgWarning)
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_fun=755, atol=1e-6, rtol=1e-7)
def test_network_flow_limited_capacity(self):
# A network flow problem with supply and demand at nodes
# and with costs and capacities along directed edges.
# http://blog.sommer-forst.de/2013/04/10/
c = [2, 2, 1, 3, 1]
bounds = [
[0, 4],
[0, 2],
[0, 2],
[0, 3],
[0, 5]]
n, p = -1, 1
A_eq = [
[n, n, 0, 0, 0],
[p, 0, n, n, 0],
[0, p, p, 0, n],
[0, 0, 0, p, p]]
b_eq = [-4, 0, 0, 4]
with suppress_warnings() as sup:
# this is an UmfpackWarning but I had trouble importing it
if has_umfpack:
sup.filter(UmfpackWarning)
sup.filter(RuntimeWarning, "scipy.linalg.solve\nIll...")
sup.filter(OptimizeWarning, "A_eq does not appear...")
sup.filter(OptimizeWarning, "Solving system with option...")
sup.filter(LinAlgWarning)
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_fun=14)
def test_simplex_algorithm_wikipedia_example(self):
# https://en.wikipedia.org/wiki/Simplex_algorithm#Example
c = [-2, -3, -4]
A_ub = [
[3, 2, 1],
[2, 5, 3]]
b_ub = [10, 15]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_fun=-20)
def test_enzo_example(self):
# https://github.com/scipy/scipy/issues/1779 lp2.py
#
# Translated from Octave code at:
# http://www.ecs.shimane-u.ac.jp/~kyoshida/lpeng.htm
# and placed under MIT licence by Enzo Michelangeli
# with permission explicitly granted by the original author,
# Prof. Kazunobu Yoshida
c = [4, 8, 3, 0, 0, 0]
A_eq = [
[2, 5, 3, -1, 0, 0],
[3, 2.5, 8, 0, -1, 0],
[8, 10, 4, 0, 0, -1]]
b_eq = [185, 155, 600]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_fun=317.5,
desired_x=[66.25, 0, 17.5, 0, 183.75, 0],
atol=6e-6, rtol=1e-7)
def test_enzo_example_b(self):
# rescued from https://github.com/scipy/scipy/pull/218
c = [2.8, 6.3, 10.8, -2.8, -6.3, -10.8]
A_eq = [[-1, -1, -1, 0, 0, 0],
[0, 0, 0, 1, 1, 1],
[1, 0, 0, 1, 0, 0],
[0, 1, 0, 0, 1, 0],
[0, 0, 1, 0, 0, 1]]
b_eq = [-0.5, 0.4, 0.3, 0.3, 0.3]
with suppress_warnings() as sup:
sup.filter(OptimizeWarning, "A_eq does not appear...")
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_fun=-1.77,
desired_x=[0.3, 0.2, 0.0, 0.0, 0.1, 0.3])
def test_enzo_example_c_with_degeneracy(self):
# rescued from https://github.com/scipy/scipy/pull/218
m = 20
c = -np.ones(m)
tmp = 2 * np.pi * np.arange(1, m + 1) / (m + 1)
A_eq = np.vstack((np.cos(tmp) - 1, np.sin(tmp)))
b_eq = [0, 0]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_fun=0, desired_x=np.zeros(m))
def test_enzo_example_c_with_unboundedness(self):
# rescued from https://github.com/scipy/scipy/pull/218
m = 50
c = -np.ones(m)
tmp = 2 * np.pi * np.arange(m) / (m + 1)
# This test relies on `cos(0) -1 == sin(0)`, so ensure that's true
# (SIMD code or -ffast-math may cause spurious failures otherwise)
row0 = np.cos(tmp) - 1
row0[0] = 0.0
row1 = np.sin(tmp)
row1[0] = 0.0
A_eq = np.vstack((row0, row1))
b_eq = [0, 0]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_unbounded(res)
def test_enzo_example_c_with_infeasibility(self):
# rescued from https://github.com/scipy/scipy/pull/218
m = 50
c = -np.ones(m)
tmp = 2 * np.pi * np.arange(m) / (m + 1)
A_eq = np.vstack((np.cos(tmp) - 1, np.sin(tmp)))
b_eq = [1, 1]
o = {key: self.options[key] for key in self.options}
o["presolve"] = False
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=o)
_assert_infeasible(res)
def test_basic_artificial_vars(self):
# Problem is chosen to test two phase simplex methods when at the end
# of phase 1 some artificial variables remain in the basis.
# Also, for `method='simplex'`, the row in the tableau corresponding
# with the artificial variables is not all zero.
c = np.array([-0.1, -0.07, 0.004, 0.004, 0.004, 0.004])
A_ub = np.array([[1.0, 0, 0, 0, 0, 0], [-1.0, 0, 0, 0, 0, 0],
[0, -1.0, 0, 0, 0, 0], [0, 1.0, 0, 0, 0, 0],
[1.0, 1.0, 0, 0, 0, 0]])
b_ub = np.array([3.0, 3.0, 3.0, 3.0, 20.0])
A_eq = np.array([[1.0, 0, -1, 1, -1, 1], [0, -1.0, -1, 1, -1, 1]])
b_eq = np.array([0, 0])
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_fun=0, desired_x=np.zeros_like(c),
atol=2e-6)
def test_optimize_result(self):
# check all fields in OptimizeResult
c, A_ub, b_ub, A_eq, b_eq, bounds = very_random_gen(0)
res = linprog(c, A_ub=A_ub, b_ub=b_ub, A_eq=A_eq, b_eq=b_eq,
bounds=bounds, method=self.method, options=self.options)
assert_(res.success)
assert_(res.nit)
assert_(not res.status)
if 'highs' not in self.method:
# HiGHS status/message tested separately
assert_(res.message == "Optimization terminated successfully.")
assert_allclose(c @ res.x, res.fun)
assert_allclose(b_eq - A_eq @ res.x, res.con, atol=1e-11)
assert_allclose(b_ub - A_ub @ res.x, res.slack, atol=1e-11)
for key in ['eqlin', 'ineqlin', 'lower', 'upper']:
if key in res.keys():
assert isinstance(res[key]['marginals'], np.ndarray)
assert isinstance(res[key]['residual'], np.ndarray)
#################
# Bug Fix Tests #
#################
def test_bug_5400(self):
# https://github.com/scipy/scipy/issues/5400
bounds = [
(0, None),
(0, 100), (0, 100), (0, 100), (0, 100), (0, 100), (0, 100),
(0, 900), (0, 900), (0, 900), (0, 900), (0, 900), (0, 900),
(0, None), (0, None), (0, None), (0, None), (0, None), (0, None)]
f = 1 / 9
g = -1e4
h = -3.1
A_ub = np.array([
[1, -2.99, 0, 0, -3, 0, 0, 0, -1, -1, 0, -1, -1, 1, 1, 0, 0, 0, 0],
[1, 0, -2.9, h, 0, -3, 0, -1, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0, 0],
[1, 0, 0, h, 0, 0, -3, -1, -1, 0, -1, -1, 0, 0, 0, 0, 0, 1, 1],
[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1],
[0, 1.99, -1, -1, 0, 0, 0, -1, f, f, 0, 0, 0, g, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 2, -1, -1, 0, 0, 0, -1, f, f, 0, g, 0, 0, 0, 0],
[0, -1, 1.9, 2.1, 0, 0, 0, f, -1, -1, 0, 0, 0, 0, 0, g, 0, 0, 0],
[0, 0, 0, 0, -1, 2, -1, 0, 0, 0, f, -1, f, 0, 0, 0, g, 0, 0],
[0, -1, -1, 2.1, 0, 0, 0, f, f, -1, 0, 0, 0, 0, 0, 0, 0, g, 0],
[0, 0, 0, 0, -1, -1, 2, 0, 0, 0, f, f, -1, 0, 0, 0, 0, 0, g]])
b_ub = np.array([
0.0, 0, 0, 100, 100, 100, 100, 100, 100, 900, 900, 900, 900, 900,
900, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0])
c = np.array([-1.0, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 0, 0, 0, 0, 0, 0])
with suppress_warnings() as sup:
sup.filter(OptimizeWarning,
"Solving system with option 'sym_pos'")
sup.filter(RuntimeWarning, "invalid value encountered")
sup.filter(LinAlgWarning)
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_fun=-106.63507541835018)
def test_bug_6139(self):
# linprog(method='simplex') fails to find a basic feasible solution
# if phase 1 pseudo-objective function is outside the provided tol.
# https://github.com/scipy/scipy/issues/6139
# Note: This is not strictly a bug as the default tolerance determines
# if a result is "close enough" to zero and should not be expected
# to work for all cases.
c = np.array([1, 1, 1])
A_eq = np.array([[1., 0., 0.], [-1000., 0., - 1000.]])
b_eq = np.array([5.00000000e+00, -1.00000000e+04])
A_ub = -np.array([[0., 1000000., 1010000.]])
b_ub = -np.array([10000000.])
bounds = (None, None)
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_fun=14.95,
desired_x=np.array([5, 4.95, 5]))
def test_bug_6690(self):
# linprog simplex used to violate bound constraint despite reporting
# success.
# https://github.com/scipy/scipy/issues/6690
A_eq = np.array([[0, 0, 0, 0.93, 0, 0.65, 0, 0, 0.83, 0]])
b_eq = np.array([0.9626])
A_ub = np.array([
[0, 0, 0, 1.18, 0, 0, 0, -0.2, 0, -0.22],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0.43, 0, 0, 0, 0, 0, 0],
[0, -1.22, -0.25, 0, 0, 0, -2.06, 0, 0, 1.37],
[0, 0, 0, 0, 0, 0, 0, -0.25, 0, 0]
])
b_ub = np.array([0.615, 0, 0.172, -0.869, -0.022])
bounds = np.array([
[-0.84, -0.97, 0.34, 0.4, -0.33, -0.74, 0.47, 0.09, -1.45, -0.73],
[0.37, 0.02, 2.86, 0.86, 1.18, 0.5, 1.76, 0.17, 0.32, -0.15]
]).T
c = np.array([
-1.64, 0.7, 1.8, -1.06, -1.16, 0.26, 2.13, 1.53, 0.66, 0.28
])
with suppress_warnings() as sup:
if has_umfpack:
sup.filter(UmfpackWarning)
sup.filter(OptimizeWarning,
"Solving system with option 'cholesky'")
sup.filter(OptimizeWarning, "Solving system with option 'sym_pos'")
sup.filter(RuntimeWarning, "invalid value encountered")
sup.filter(LinAlgWarning)
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
desired_fun = -1.19099999999
desired_x = np.array([0.3700, -0.9700, 0.3400, 0.4000, 1.1800,
0.5000, 0.4700, 0.0900, 0.3200, -0.7300])
_assert_success(res, desired_fun=desired_fun, desired_x=desired_x)
# Add small tol value to ensure arrays are less than or equal.
atol = 1e-6
assert_array_less(bounds[:, 0] - atol, res.x)
assert_array_less(res.x, bounds[:, 1] + atol)
def test_bug_7044(self):
# linprog simplex failed to "identify correct constraints" (?)
# leading to a non-optimal solution if A is rank-deficient.
# https://github.com/scipy/scipy/issues/7044
A_eq, b_eq, c, _, _ = magic_square(3)
with suppress_warnings() as sup:
sup.filter(OptimizeWarning, "A_eq does not appear...")
sup.filter(RuntimeWarning, "invalid value encountered")
sup.filter(LinAlgWarning)
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
desired_fun = 1.730550597
_assert_success(res, desired_fun=desired_fun)
assert_allclose(A_eq.dot(res.x), b_eq)
assert_array_less(np.zeros(res.x.size) - 1e-5, res.x)
def test_bug_7237(self):
# https://github.com/scipy/scipy/issues/7237
# linprog simplex "explodes" when the pivot value is very
# close to zero.
c = np.array([-1, 0, 0, 0, 0, 0, 0, 0, 0])
A_ub = np.array([
[1., -724., 911., -551., -555., -896., 478., -80., -293.],
[1., 566., 42., 937., 233., 883., 392., -909., 57.],
[1., -208., -894., 539., 321., 532., -924., 942., 55.],
[1., 857., -859., 83., 462., -265., -971., 826., 482.],
[1., 314., -424., 245., -424., 194., -443., -104., -429.],
[1., 540., 679., 361., 149., -827., 876., 633., 302.],
[0., -1., -0., -0., -0., -0., -0., -0., -0.],
[0., -0., -1., -0., -0., -0., -0., -0., -0.],
[0., -0., -0., -1., -0., -0., -0., -0., -0.],
[0., -0., -0., -0., -1., -0., -0., -0., -0.],
[0., -0., -0., -0., -0., -1., -0., -0., -0.],
[0., -0., -0., -0., -0., -0., -1., -0., -0.],
[0., -0., -0., -0., -0., -0., -0., -1., -0.],
[0., -0., -0., -0., -0., -0., -0., -0., -1.],
[0., 1., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 1., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 1., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 1., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 1., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 1., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 1., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 1.]
])
b_ub = np.array([
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 1., 1., 1., 1., 1., 1., 1., 1.])
A_eq = np.array([[0., 1., 1., 1., 1., 1., 1., 1., 1.]])
b_eq = np.array([[1.]])
bounds = [(None, None)] * 9
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_fun=108.568535, atol=1e-6)
def test_bug_8174(self):
# https://github.com/scipy/scipy/issues/8174
# The simplex method sometimes "explodes" if the pivot value is very
# close to zero.
A_ub = np.array([
[22714, 1008, 13380, -2713.5, -1116],
[-4986, -1092, -31220, 17386.5, 684],
[-4986, 0, 0, -2713.5, 0],
[22714, 0, 0, 17386.5, 0]])
b_ub = np.zeros(A_ub.shape[0])
c = -np.ones(A_ub.shape[1])
bounds = [(0, 1)] * A_ub.shape[1]
with suppress_warnings() as sup:
sup.filter(RuntimeWarning, "invalid value encountered")
sup.filter(LinAlgWarning)
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
if self.options.get('tol', 1e-9) < 1e-10 and self.method == 'simplex':
_assert_unable_to_find_basic_feasible_sol(res)
else:
_assert_success(res, desired_fun=-2.0080717488789235, atol=1e-6)
def test_bug_8174_2(self):
# Test supplementary example from issue 8174.
# https://github.com/scipy/scipy/issues/8174
# https://stackoverflow.com/questions/47717012/linprog-in-scipy-optimize-checking-solution
c = np.array([1, 0, 0, 0, 0, 0, 0])
A_ub = -np.identity(7)
b_ub = np.array([[-2], [-2], [-2], [-2], [-2], [-2], [-2]])
A_eq = np.array([
[1, 1, 1, 1, 1, 1, 0],
[0.3, 1.3, 0.9, 0, 0, 0, -1],
[0.3, 0, 0, 0, 0, 0, -2/3],
[0, 0.65, 0, 0, 0, 0, -1/15],
[0, 0, 0.3, 0, 0, 0, -1/15]
])
b_eq = np.array([[100], [0], [0], [0], [0]])
with suppress_warnings() as sup:
if has_umfpack:
sup.filter(UmfpackWarning)
sup.filter(OptimizeWarning, "A_eq does not appear...")
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_fun=43.3333333331385)
def test_bug_8561(self):
# Test that pivot row is chosen correctly when using Bland's rule
# This was originally written for the simplex method with
# Bland's rule only, but it doesn't hurt to test all methods/options
# https://github.com/scipy/scipy/issues/8561
c = np.array([7, 0, -4, 1.5, 1.5])
A_ub = np.array([
[4, 5.5, 1.5, 1.0, -3.5],
[1, -2.5, -2, 2.5, 0.5],
[3, -0.5, 4, -12.5, -7],
[-1, 4.5, 2, -3.5, -2],
[5.5, 2, -4.5, -1, 9.5]])
b_ub = np.array([0, 0, 0, 0, 1])
res = linprog(c, A_ub=A_ub, b_ub=b_ub, options=self.options,
method=self.method)
_assert_success(res, desired_x=[0, 0, 19, 16/3, 29/3])
def test_bug_8662(self):
# linprog simplex used to report incorrect optimal results
# https://github.com/scipy/scipy/issues/8662
c = [-10, 10, 6, 3]
A_ub = [[8, -8, -4, 6],
[-8, 8, 4, -6],
[-4, 4, 8, -4],
[3, -3, -3, -10]]
b_ub = [9, -9, -9, -4]
bounds = [(0, None), (0, None), (0, None), (0, None)]
desired_fun = 36.0000000000
with suppress_warnings() as sup:
if has_umfpack:
sup.filter(UmfpackWarning)
sup.filter(RuntimeWarning, "invalid value encountered")
sup.filter(LinAlgWarning)
res1 = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
# Set boundary condition as a constraint
A_ub.append([0, 0, -1, 0])
b_ub.append(0)
bounds[2] = (None, None)
with suppress_warnings() as sup:
if has_umfpack:
sup.filter(UmfpackWarning)
sup.filter(RuntimeWarning, "invalid value encountered")
sup.filter(LinAlgWarning)
res2 = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
rtol = 1e-5
_assert_success(res1, desired_fun=desired_fun, rtol=rtol)
_assert_success(res2, desired_fun=desired_fun, rtol=rtol)
def test_bug_8663(self):
# exposed a bug in presolve
# https://github.com/scipy/scipy/issues/8663
c = [1, 5]
A_eq = [[0, -7]]
b_eq = [-6]
bounds = [(0, None), (None, None)]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_x=[0, 6./7], desired_fun=5*6./7)
def test_bug_8664(self):
# interior-point has trouble with this when presolve is off
# tested for interior-point with presolve off in TestLinprogIPSpecific
# https://github.com/scipy/scipy/issues/8664
c = [4]
A_ub = [[2], [5]]
b_ub = [4, 4]
A_eq = [[0], [-8], [9]]
b_eq = [3, 2, 10]
with suppress_warnings() as sup:
sup.filter(RuntimeWarning)
sup.filter(OptimizeWarning, "Solving system with option...")
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_infeasible(res)
def test_bug_8973(self):
"""
Test whether bug described at:
https://github.com/scipy/scipy/issues/8973
was fixed.
"""
c = np.array([0, 0, 0, 1, -1])
A_ub = np.array([[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]])
b_ub = np.array([2, -2])
bounds = [(None, None), (None, None), (None, None), (-1, 1), (-1, 1)]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
# solution vector x is not unique
_assert_success(res, desired_fun=-2)
# HiGHS IPM had an issue where the following wasn't true!
assert_equal(c @ res.x, res.fun)
def test_bug_8973_2(self):
"""
Additional test for:
https://github.com/scipy/scipy/issues/8973
suggested in
https://github.com/scipy/scipy/pull/8985
review by @antonior92
"""
c = np.zeros(1)
A_ub = np.array([[1]])
b_ub = np.array([-2])
bounds = (None, None)
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_x=[-2], desired_fun=0)
def test_bug_10124(self):
"""
Test for linprog docstring problem
'disp'=True caused revised simplex failure
"""
c = np.zeros(1)
A_ub = np.array([[1]])
b_ub = np.array([-2])
bounds = (None, None)
c = [-1, 4]
A_ub = [[-3, 1], [1, 2]]
b_ub = [6, 4]
bounds = [(None, None), (-3, None)]
o = {"disp": True}
o.update(self.options)
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=o)
_assert_success(res, desired_x=[10, -3], desired_fun=-22)
def test_bug_10349(self):
"""
Test for redundancy removal tolerance issue
https://github.com/scipy/scipy/issues/10349
"""
A_eq = np.array([[1, 1, 0, 0, 0, 0],
[0, 0, 1, 1, 0, 0],
[0, 0, 0, 0, 1, 1],
[1, 0, 1, 0, 0, 0],
[0, 0, 0, 1, 1, 0],
[0, 1, 0, 0, 0, 1]])
b_eq = np.array([221, 210, 10, 141, 198, 102])
c = np.concatenate((0, 1, np.zeros(4)), axis=None)
with suppress_warnings() as sup:
sup.filter(OptimizeWarning, "A_eq does not appear...")
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options)
_assert_success(res, desired_x=[129, 92, 12, 198, 0, 10], desired_fun=92)
@pytest.mark.skipif(sys.platform == 'darwin',
reason=("Failing on some local macOS builds, "
"see gh-13846"))
def test_bug_10466(self):
"""
Test that autoscale fixes poorly-scaled problem
"""
c = [-8., -0., -8., -0., -8., -0., -0., -0., -0., -0., -0., -0., -0.]
A_eq = [[1., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 1., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 1., 1., 0., 0., 0., 0., 0., 0., 0.],
[1., 0., 1., 0., 1., 0., -1., 0., 0., 0., 0., 0., 0.],
[1., 0., 1., 0., 1., 0., 0., 1., 0., 0., 0., 0., 0.],
[1., 0., 0., 0., 0., 0., 0., 0., 1., 0., 0., 0., 0.],
[1., 0., 0., 0., 0., 0., 0., 0., 0., 1., 0., 0., 0.],
[1., 0., 1., 0., 1., 0., 0., 0., 0., 0., 1., 0., 0.],
[0., 0., 1., 0., 1., 0., 0., 0., 0., 0., 0., 1., 0.],
[0., 0., 1., 0., 1., 0., 0., 0., 0., 0., 0., 0., 1.]]
b_eq = [3.14572800e+08, 4.19430400e+08, 5.24288000e+08,
1.00663296e+09, 1.07374182e+09, 1.07374182e+09,
1.07374182e+09, 1.07374182e+09, 1.07374182e+09,
1.07374182e+09]
o = {}
# HiGHS methods don't use autoscale option
if not self.method.startswith("highs"):
o = {"autoscale": True}
o.update(self.options)
with suppress_warnings() as sup:
sup.filter(OptimizeWarning, "Solving system with option...")
if has_umfpack:
sup.filter(UmfpackWarning)
sup.filter(RuntimeWarning, "scipy.linalg.solve\nIll...")
sup.filter(RuntimeWarning, "divide by zero encountered...")
sup.filter(RuntimeWarning, "overflow encountered...")
sup.filter(RuntimeWarning, "invalid value encountered...")
sup.filter(LinAlgWarning, "Ill-conditioned matrix...")
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=o)
assert_allclose(res.fun, -8589934560)
def test_bug_20584(self):
"""
Test that when integrality is a list of all zeros, linprog gives the
same result as when it is an array of all zeros / integrality=None
"""
c = [1, 1]
A_ub = [[-1, 0]]
b_ub = [-2.5]
res1 = linprog(c, A_ub=A_ub, b_ub=b_ub, integrality=[0, 0])
res2 = linprog(c, A_ub=A_ub, b_ub=b_ub, integrality=np.asarray([0, 0]))
res3 = linprog(c, A_ub=A_ub, b_ub=b_ub, integrality=None)
assert_equal(res1.x, res2.x)
assert_equal(res1.x, res3.x)
#########################
# Method-specific Tests #
#########################
@pytest.mark.filterwarnings("ignore::DeprecationWarning")
class LinprogSimplexTests(LinprogCommonTests):
method = "simplex"
@pytest.mark.filterwarnings("ignore::DeprecationWarning")
class LinprogIPTests(LinprogCommonTests):
method = "interior-point"
def test_bug_10466(self):
pytest.skip("Test is failing, but solver is deprecated.")
@pytest.mark.filterwarnings("ignore::DeprecationWarning")
class LinprogRSTests(LinprogCommonTests):
method = "revised simplex"
# Revised simplex does not reliably solve these problems.
# Failure is intermittent due to the random choice of elements to complete
# the basis after phase 1 terminates. In any case, linprog exists
# gracefully, reporting numerical difficulties. I do not think this should
# prevent revised simplex from being merged, as it solves the problems
# most of the time and solves a broader range of problems than the existing
# simplex implementation.
# I believe that the root cause is the same for all three and that this
# same issue prevents revised simplex from solving many other problems
# reliably. Somehow the pivoting rule allows the algorithm to pivot into
# a singular basis. I haven't been able to find a reference that
# acknowledges this possibility, suggesting that there is a bug. On the
# other hand, the pivoting rule is quite simple, and I can't find a
# mistake, which suggests that this is a possibility with the pivoting
# rule. Hopefully, a better pivoting rule will fix the issue.
def test_bug_5400(self):
pytest.skip("Intermittent failure acceptable.")
def test_bug_8662(self):
pytest.skip("Intermittent failure acceptable.")
def test_network_flow(self):
pytest.skip("Intermittent failure acceptable.")
class LinprogHiGHSTests(LinprogCommonTests):
def test_callback(self):
# this is the problem from test_callback
def cb(res):
return None
c = np.array([-3, -2])
A_ub = [[2, 1], [1, 1], [1, 0]]
b_ub = [10, 8, 4]
assert_raises(NotImplementedError, linprog, c, A_ub=A_ub, b_ub=b_ub,
callback=cb, method=self.method)
res = linprog(c, A_ub=A_ub, b_ub=b_ub, method=self.method)
_assert_success(res, desired_fun=-18.0, desired_x=[2, 6])
@pytest.mark.parametrize("options",
[{"maxiter": -1},
{"disp": -1},
{"presolve": -1},
{"time_limit": -1},
{"dual_feasibility_tolerance": -1},
{"primal_feasibility_tolerance": -1},
{"ipm_optimality_tolerance": -1},
{"simplex_dual_edge_weight_strategy": "ekki"},
])
def test_invalid_option_values(self, options):
def f(options):
linprog(1, method=self.method, options=options)
options.update(self.options)
assert_warns(OptimizeWarning, f, options=options)
def test_crossover(self):
A_eq, b_eq, c, _, _ = magic_square(4)
bounds = (0, 1)
res = linprog(c, A_eq=A_eq, b_eq=b_eq,
bounds=bounds, method=self.method, options=self.options)
# there should be nonzero crossover iterations for IPM (only)
assert_equal(res.crossover_nit == 0, self.method != "highs-ipm")
@pytest.mark.fail_slow(5)
def test_marginals(self):
# Ensure lagrange multipliers are correct by comparing the derivative
# w.r.t. b_ub/b_eq/ub/lb to the reported duals.
c, A_ub, b_ub, A_eq, b_eq, bounds = very_random_gen(seed=0)
res = linprog(c, A_ub=A_ub, b_ub=b_ub, A_eq=A_eq, b_eq=b_eq,
bounds=bounds, method=self.method, options=self.options)
lb, ub = bounds.T
# sensitivity w.r.t. b_ub
def f_bub(x):
return linprog(c, A_ub, x, A_eq, b_eq, bounds,
method=self.method).fun
dfdbub = approx_derivative(f_bub, b_ub, method='3-point', f0=res.fun)
assert_allclose(res.ineqlin.marginals, dfdbub)
# sensitivity w.r.t. b_eq
def f_beq(x):
return linprog(c, A_ub, b_ub, A_eq, x, bounds,
method=self.method).fun
dfdbeq = approx_derivative(f_beq, b_eq, method='3-point', f0=res.fun)
assert_allclose(res.eqlin.marginals, dfdbeq)
# sensitivity w.r.t. lb
def f_lb(x):
bounds = np.array([x, ub]).T
return linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method).fun
with np.errstate(invalid='ignore'):
# approx_derivative has trouble where lb is infinite
dfdlb = approx_derivative(f_lb, lb, method='3-point', f0=res.fun)
dfdlb[~np.isfinite(lb)] = 0
assert_allclose(res.lower.marginals, dfdlb)
# sensitivity w.r.t. ub
def f_ub(x):
bounds = np.array([lb, x]).T
return linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method).fun
with np.errstate(invalid='ignore'):
dfdub = approx_derivative(f_ub, ub, method='3-point', f0=res.fun)
dfdub[~np.isfinite(ub)] = 0
assert_allclose(res.upper.marginals, dfdub)
def test_dual_feasibility(self):
# Ensure solution is dual feasible using marginals
c, A_ub, b_ub, A_eq, b_eq, bounds = very_random_gen(seed=42)
res = linprog(c, A_ub=A_ub, b_ub=b_ub, A_eq=A_eq, b_eq=b_eq,
bounds=bounds, method=self.method, options=self.options)
# KKT dual feasibility equation from Theorem 1 from
# http://www.personal.psu.edu/cxg286/LPKKT.pdf
resid = (-c + A_ub.T @ res.ineqlin.marginals +
A_eq.T @ res.eqlin.marginals +
res.upper.marginals +
res.lower.marginals)
assert_allclose(resid, 0, atol=1e-12)
def test_complementary_slackness(self):
# Ensure that the complementary slackness condition is satisfied.
c, A_ub, b_ub, A_eq, b_eq, bounds = very_random_gen(seed=42)
res = linprog(c, A_ub=A_ub, b_ub=b_ub, A_eq=A_eq, b_eq=b_eq,
bounds=bounds, method=self.method, options=self.options)
# KKT complementary slackness equation from Theorem 1 from
# http://www.personal.psu.edu/cxg286/LPKKT.pdf modified for
# non-zero RHS
assert np.allclose(res.ineqlin.marginals @ (b_ub - A_ub @ res.x), 0)
################################
# Simplex Option-Specific Tests#
################################
class TestLinprogSimplexDefault(LinprogSimplexTests):
def setup_method(self):
self.options = {}
def test_bug_5400(self):
pytest.skip("Simplex fails on this problem.")
def test_bug_7237_low_tol(self):
# Fails if the tolerance is too strict. Here, we test that
# even if the solution is wrong, the appropriate error is raised.
pytest.skip("Simplex fails on this problem.")
def test_bug_8174_low_tol(self):
# Fails if the tolerance is too strict. Here, we test that
# even if the solution is wrong, the appropriate warning is issued.
self.options.update({'tol': 1e-12})
with pytest.warns(OptimizeWarning):
super().test_bug_8174()
class TestLinprogSimplexBland(LinprogSimplexTests):
def setup_method(self):
self.options = {'bland': True}
def test_bug_5400(self):
pytest.skip("Simplex fails on this problem.")
def test_bug_8174_low_tol(self):
# Fails if the tolerance is too strict. Here, we test that
# even if the solution is wrong, the appropriate error is raised.
self.options.update({'tol': 1e-12})
with pytest.raises(AssertionError):
with pytest.warns(OptimizeWarning):
super().test_bug_8174()
class TestLinprogSimplexNoPresolve(LinprogSimplexTests):
def setup_method(self):
self.options = {'presolve': False}
is_32_bit = np.intp(0).itemsize < 8
is_linux = sys.platform.startswith('linux')
@pytest.mark.xfail(
condition=is_32_bit and is_linux,
reason='Fails with warning on 32-bit linux')
def test_bug_5400(self):
super().test_bug_5400()
def test_bug_6139_low_tol(self):
# Linprog(method='simplex') fails to find a basic feasible solution
# if phase 1 pseudo-objective function is outside the provided tol.
# https://github.com/scipy/scipy/issues/6139
# Without ``presolve`` eliminating such rows the result is incorrect.
self.options.update({'tol': 1e-12})
with pytest.raises(AssertionError, match='linprog status 4'):
return super().test_bug_6139()
def test_bug_7237_low_tol(self):
pytest.skip("Simplex fails on this problem.")
def test_bug_8174_low_tol(self):
# Fails if the tolerance is too strict. Here, we test that
# even if the solution is wrong, the appropriate warning is issued.
self.options.update({'tol': 1e-12})
with pytest.warns(OptimizeWarning):
super().test_bug_8174()
def test_unbounded_no_nontrivial_constraints_1(self):
pytest.skip("Tests behavior specific to presolve")
def test_unbounded_no_nontrivial_constraints_2(self):
pytest.skip("Tests behavior specific to presolve")
#######################################
# Interior-Point Option-Specific Tests#
#######################################
class TestLinprogIPDense(LinprogIPTests):
options = {"sparse": False}
# see https://github.com/scipy/scipy/issues/20216 for skip reason
@pytest.mark.skipif(
sys.platform == 'darwin',
reason="Fails on some macOS builds for reason not relevant to test"
)
def test_bug_6139(self):
super().test_bug_6139()
if has_cholmod:
class TestLinprogIPSparseCholmod(LinprogIPTests):
options = {"sparse": True, "cholesky": True}
if has_umfpack:
class TestLinprogIPSparseUmfpack(LinprogIPTests):
options = {"sparse": True, "cholesky": False}
def test_network_flow_limited_capacity(self):
pytest.skip("Failing due to numerical issues on some platforms.")
class TestLinprogIPSparse(LinprogIPTests):
options = {"sparse": True, "cholesky": False, "sym_pos": False}
@pytest.mark.skipif(
sys.platform == 'darwin',
reason="Fails on macOS x86 Accelerate builds (gh-20510)"
)
@pytest.mark.xfail_on_32bit("This test is sensitive to machine epsilon level "
"perturbations in linear system solution in "
"_linprog_ip._sym_solve.")
def test_bug_6139(self):
super().test_bug_6139()
@pytest.mark.xfail(reason='Fails with ATLAS, see gh-7877')
def test_bug_6690(self):
# Test defined in base class, but can't mark as xfail there
super().test_bug_6690()
def test_magic_square_sparse_no_presolve(self):
# test linprog with a problem with a rank-deficient A_eq matrix
A_eq, b_eq, c, _, _ = magic_square(3)
bounds = (0, 1)
with suppress_warnings() as sup:
if has_umfpack:
sup.filter(UmfpackWarning)
sup.filter(MatrixRankWarning, "Matrix is exactly singular")
sup.filter(OptimizeWarning, "Solving system with option...")
o = {key: self.options[key] for key in self.options}
o["presolve"] = False
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=o)
_assert_success(res, desired_fun=1.730550597)
def test_sparse_solve_options(self):
# checking that problem is solved with all column permutation options
A_eq, b_eq, c, _, _ = magic_square(3)
with suppress_warnings() as sup:
sup.filter(OptimizeWarning, "A_eq does not appear...")
sup.filter(OptimizeWarning, "Invalid permc_spec option")
o = {key: self.options[key] for key in self.options}
permc_specs = ('NATURAL', 'MMD_ATA', 'MMD_AT_PLUS_A',
'COLAMD', 'ekki-ekki-ekki')
# 'ekki-ekki-ekki' raises warning about invalid permc_spec option
# and uses default
for permc_spec in permc_specs:
o["permc_spec"] = permc_spec
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=o)
_assert_success(res, desired_fun=1.730550597)
class TestLinprogIPSparsePresolve(LinprogIPTests):
options = {"sparse": True, "_sparse_presolve": True}
@pytest.mark.skipif(
sys.platform == 'darwin',
reason="Fails on macOS x86 Accelerate builds (gh-20510)"
)
@pytest.mark.xfail_on_32bit("This test is sensitive to machine epsilon level "
"perturbations in linear system solution in "
"_linprog_ip._sym_solve.")
def test_bug_6139(self):
super().test_bug_6139()
def test_enzo_example_c_with_infeasibility(self):
pytest.skip('_sparse_presolve=True incompatible with presolve=False')
@pytest.mark.xfail(reason='Fails with ATLAS, see gh-7877')
def test_bug_6690(self):
# Test defined in base class, but can't mark as xfail there
super().test_bug_6690()
@pytest.mark.filterwarnings("ignore::DeprecationWarning")
class TestLinprogIPSpecific:
method = "interior-point"
# the following tests don't need to be performed separately for
# sparse presolve, sparse after presolve, and dense
def test_solver_select(self):
# check that default solver is selected as expected
if has_cholmod:
options = {'sparse': True, 'cholesky': True}
elif has_umfpack:
options = {'sparse': True, 'cholesky': False}
else:
options = {'sparse': True, 'cholesky': False, 'sym_pos': False}
A, b, c = lpgen_2d(20, 20)
res1 = linprog(c, A_ub=A, b_ub=b, method=self.method, options=options)
res2 = linprog(c, A_ub=A, b_ub=b, method=self.method) # default solver
assert_allclose(res1.fun, res2.fun,
err_msg="linprog default solver unexpected result",
rtol=2e-15, atol=1e-15)
def test_unbounded_below_no_presolve_original(self):
# formerly caused segfault in TravisCI w/ "cholesky":True
c = [-1]
bounds = [(None, 1)]
res = linprog(c=c, bounds=bounds,
method=self.method,
options={"presolve": False, "cholesky": True})
_assert_success(res, desired_fun=-1)
def test_cholesky(self):
# use cholesky factorization and triangular solves
A, b, c = lpgen_2d(20, 20)
res = linprog(c, A_ub=A, b_ub=b, method=self.method,
options={"cholesky": True}) # only for dense
_assert_success(res, desired_fun=-64.049494229)
def test_alternate_initial_point(self):
# use "improved" initial point
A, b, c = lpgen_2d(20, 20)
with suppress_warnings() as sup:
sup.filter(RuntimeWarning, "scipy.linalg.solve\nIll...")
sup.filter(OptimizeWarning, "Solving system with option...")
sup.filter(LinAlgWarning, "Ill-conditioned matrix...")
res = linprog(c, A_ub=A, b_ub=b, method=self.method,
options={"ip": True, "disp": True})
# ip code is independent of sparse/dense
_assert_success(res, desired_fun=-64.049494229)
def test_bug_8664(self):
# interior-point has trouble with this when presolve is off
c = [4]
A_ub = [[2], [5]]
b_ub = [4, 4]
A_eq = [[0], [-8], [9]]
b_eq = [3, 2, 10]
with suppress_warnings() as sup:
sup.filter(RuntimeWarning)
sup.filter(OptimizeWarning, "Solving system with option...")
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options={"presolve": False})
assert_(not res.success, "Incorrectly reported success")
########################################
# Revised Simplex Option-Specific Tests#
########################################
class TestLinprogRSCommon(LinprogRSTests):
options = {}
def test_cyclic_bland(self):
pytest.skip("Intermittent failure acceptable.")
def test_nontrivial_problem_with_guess(self):
c, A_ub, b_ub, A_eq, b_eq, x_star, f_star = nontrivial_problem()
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options, x0=x_star)
_assert_success(res, desired_fun=f_star, desired_x=x_star)
assert_equal(res.nit, 0)
def test_nontrivial_problem_with_unbounded_variables(self):
c, A_ub, b_ub, A_eq, b_eq, x_star, f_star = nontrivial_problem()
bounds = [(None, None), (None, None), (0, None), (None, None)]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options, x0=x_star)
_assert_success(res, desired_fun=f_star, desired_x=x_star)
assert_equal(res.nit, 0)
def test_nontrivial_problem_with_bounded_variables(self):
c, A_ub, b_ub, A_eq, b_eq, x_star, f_star = nontrivial_problem()
bounds = [(None, 1), (1, None), (0, None), (.4, .6)]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options, x0=x_star)
_assert_success(res, desired_fun=f_star, desired_x=x_star)
assert_equal(res.nit, 0)
def test_nontrivial_problem_with_negative_unbounded_variable(self):
c, A_ub, b_ub, A_eq, b_eq, x_star, f_star = nontrivial_problem()
b_eq = [4]
x_star = np.array([-219/385, 582/385, 0, 4/10])
f_star = 3951/385
bounds = [(None, None), (1, None), (0, None), (.4, .6)]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options, x0=x_star)
_assert_success(res, desired_fun=f_star, desired_x=x_star)
assert_equal(res.nit, 0)
def test_nontrivial_problem_with_bad_guess(self):
c, A_ub, b_ub, A_eq, b_eq, x_star, f_star = nontrivial_problem()
bad_guess = [1, 2, 3, .5]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options, x0=bad_guess)
assert_equal(res.status, 6)
def test_redundant_constraints_with_guess(self):
A, b, c, _, _ = magic_square(3)
p = np.random.rand(*c.shape)
with suppress_warnings() as sup:
sup.filter(OptimizeWarning, "A_eq does not appear...")
sup.filter(RuntimeWarning, "invalid value encountered")
sup.filter(LinAlgWarning)
res = linprog(c, A_eq=A, b_eq=b, method=self.method)
res2 = linprog(c, A_eq=A, b_eq=b, method=self.method, x0=res.x)
res3 = linprog(c + p, A_eq=A, b_eq=b, method=self.method, x0=res.x)
_assert_success(res2, desired_fun=1.730550597)
assert_equal(res2.nit, 0)
_assert_success(res3)
assert_(res3.nit < res.nit) # hot start reduces iterations
class TestLinprogRSBland(LinprogRSTests):
options = {"pivot": "bland"}
############################################
# HiGHS-Simplex-Dual Option-Specific Tests #
############################################
class TestLinprogHiGHSSimplexDual(LinprogHiGHSTests):
method = "highs-ds"
options = {}
def test_lad_regression(self):
'''
The scaled model should be optimal, i.e. not produce unscaled model
infeasible. See https://github.com/ERGO-Code/HiGHS/issues/494.
'''
# Test to ensure gh-13610 is resolved (mismatch between HiGHS scaled
# and unscaled model statuses)
c, A_ub, b_ub, bnds = l1_regression_prob()
res = linprog(c, A_ub=A_ub, b_ub=b_ub, bounds=bnds,
method=self.method, options=self.options)
assert_equal(res.status, 0)
assert_(res.x is not None)
assert_(np.all(res.slack > -1e-6))
assert_(np.all(res.x <= [np.inf if ub is None else ub
for lb, ub in bnds]))
assert_(np.all(res.x >= [-np.inf if lb is None else lb - 1e-7
for lb, ub in bnds]))
###################################
# HiGHS-IPM Option-Specific Tests #
###################################
class TestLinprogHiGHSIPM(LinprogHiGHSTests):
method = "highs-ipm"
options = {}
###################################
# HiGHS-MIP Option-Specific Tests #
###################################
class TestLinprogHiGHSMIP:
method = "highs"
options = {}
@pytest.mark.fail_slow(5)
@pytest.mark.xfail(condition=(sys.maxsize < 2 ** 32 and
platform.system() == "Linux"),
run=False,
reason="gh-16347")
def test_mip1(self):
# solve non-relaxed magic square problem (finally!)
# also check that values are all integers - they don't always
# come out of HiGHS that way
n = 4
A, b, c, numbers, M = magic_square(n)
bounds = [(0, 1)] * len(c)
integrality = [1] * len(c)
res = linprog(c=c*0, A_eq=A, b_eq=b, bounds=bounds,
method=self.method, integrality=integrality)
s = (numbers.flatten() * res.x).reshape(n**2, n, n)
square = np.sum(s, axis=0)
np.testing.assert_allclose(square.sum(axis=0), M)
np.testing.assert_allclose(square.sum(axis=1), M)
np.testing.assert_allclose(np.diag(square).sum(), M)
np.testing.assert_allclose(np.diag(square[:, ::-1]).sum(), M)
np.testing.assert_allclose(res.x, np.round(res.x), atol=1e-12)
def test_mip2(self):
# solve MIP with inequality constraints and all integer constraints
# source: slide 5,
# https://www.cs.upc.edu/~erodri/webpage/cps/theory/lp/milp/slides.pdf
# use all array inputs to test gh-16681 (integrality couldn't be array)
A_ub = np.array([[2, -2], [-8, 10]])
b_ub = np.array([-1, 13])
c = -np.array([1, 1])
bounds = np.array([(0, np.inf)] * len(c))
integrality = np.ones_like(c)
res = linprog(c=c, A_ub=A_ub, b_ub=b_ub, bounds=bounds,
method=self.method, integrality=integrality)
np.testing.assert_allclose(res.x, [1, 2])
np.testing.assert_allclose(res.fun, -3)
def test_mip3(self):
# solve MIP with inequality constraints and all integer constraints
# source: https://en.wikipedia.org/wiki/Integer_programming#Example
A_ub = np.array([[-1, 1], [3, 2], [2, 3]])
b_ub = np.array([1, 12, 12])
c = -np.array([0, 1])
bounds = [(0, np.inf)] * len(c)
integrality = [1] * len(c)
res = linprog(c=c, A_ub=A_ub, b_ub=b_ub, bounds=bounds,
method=self.method, integrality=integrality)
np.testing.assert_allclose(res.fun, -2)
# two optimal solutions possible, just need one of them
assert np.allclose(res.x, [1, 2]) or np.allclose(res.x, [2, 2])
def test_mip4(self):
# solve MIP with inequality constraints and only one integer constraint
# source: https://www.mathworks.com/help/optim/ug/intlinprog.html
A_ub = np.array([[-1, -2], [-4, -1], [2, 1]])
b_ub = np.array([14, -33, 20])
c = np.array([8, 1])
bounds = [(0, np.inf)] * len(c)
integrality = [0, 1]
res = linprog(c=c, A_ub=A_ub, b_ub=b_ub, bounds=bounds,
method=self.method, integrality=integrality)
np.testing.assert_allclose(res.x, [6.5, 7])
np.testing.assert_allclose(res.fun, 59)
def test_mip5(self):
# solve MIP with inequality and inequality constraints
# source: https://www.mathworks.com/help/optim/ug/intlinprog.html
A_ub = np.array([[1, 1, 1]])
b_ub = np.array([7])
A_eq = np.array([[4, 2, 1]])
b_eq = np.array([12])
c = np.array([-3, -2, -1])
bounds = [(0, np.inf), (0, np.inf), (0, 1)]
integrality = [0, 1, 0]
res = linprog(c=c, A_ub=A_ub, b_ub=b_ub, A_eq=A_eq, b_eq=b_eq,
bounds=bounds, method=self.method,
integrality=integrality)
np.testing.assert_allclose(res.x, [0, 6, 0])
np.testing.assert_allclose(res.fun, -12)
# gh-16897: these fields were not present, ensure that they are now
assert res.get("mip_node_count", None) is not None
assert res.get("mip_dual_bound", None) is not None
assert res.get("mip_gap", None) is not None
@pytest.mark.slow
@pytest.mark.timeout(120) # prerelease_deps_coverage_64bit_blas job
def test_mip6(self):
# solve a larger MIP with only equality constraints
# source: https://www.mathworks.com/help/optim/ug/intlinprog.html
A_eq = np.array([[22, 13, 26, 33, 21, 3, 14, 26],
[39, 16, 22, 28, 26, 30, 23, 24],
[18, 14, 29, 27, 30, 38, 26, 26],
[41, 26, 28, 36, 18, 38, 16, 26]])
b_eq = np.array([7872, 10466, 11322, 12058])
c = np.array([2, 10, 13, 17, 7, 5, 7, 3])
bounds = [(0, np.inf)]*8
integrality = [1]*8
res = linprog(c=c, A_eq=A_eq, b_eq=b_eq, bounds=bounds,
method=self.method, integrality=integrality)
np.testing.assert_allclose(res.fun, 1854)
@pytest.mark.xslow
def test_mip_rel_gap_passdown(self):
# MIP taken from test_mip6, solved with different values of mip_rel_gap
# solve a larger MIP with only equality constraints
# source: https://www.mathworks.com/help/optim/ug/intlinprog.html
A_eq = np.array([[22, 13, 26, 33, 21, 3, 14, 26],
[39, 16, 22, 28, 26, 30, 23, 24],
[18, 14, 29, 27, 30, 38, 26, 26],
[41, 26, 28, 36, 18, 38, 16, 26]])
b_eq = np.array([7872, 10466, 11322, 12058])
c = np.array([2, 10, 13, 17, 7, 5, 7, 3])
bounds = [(0, np.inf)]*8
integrality = [1]*8
mip_rel_gaps = [0.5, 0.25, 0.01, 0.001]
sol_mip_gaps = []
for mip_rel_gap in mip_rel_gaps:
res = linprog(c=c, A_ub=A_ub, b_ub=b_ub, A_eq=A_eq, b_eq=b_eq,
bounds=bounds, method=self.method,
integrality=integrality,
options={"mip_rel_gap": mip_rel_gap})
final_mip_gap = res["mip_gap"]
# assert that the solution actually has mip_gap lower than the
# required mip_rel_gap supplied
assert final_mip_gap <= mip_rel_gap
sol_mip_gaps.append(final_mip_gap)
# make sure that the mip_rel_gap parameter is actually doing something
# check that differences between solution gaps are declining
# monotonically with the mip_rel_gap parameter. np.diff does
# x[i+1] - x[i], so flip the array before differencing to get
# what should be a positive, monotone decreasing series of solution
# gaps
gap_diffs = np.diff(np.flip(sol_mip_gaps))
assert np.all(gap_diffs >= 0)
assert not np.all(gap_diffs == 0)
def test_semi_continuous(self):
# See issue #18106. This tests whether the solution is being
# checked correctly (status is 0) when integrality > 1:
# values are allowed to be 0 even if 0 is out of bounds.
c = np.array([1., 1., -1, -1])
bounds = np.array([[0.5, 1.5], [0.5, 1.5], [0.5, 1.5], [0.5, 1.5]])
integrality = np.array([2, 3, 2, 3])
res = linprog(c, bounds=bounds,
integrality=integrality, method='highs')
np.testing.assert_allclose(res.x, [0, 0, 1.5, 1])
assert res.status == 0
###########################
# Autoscale-Specific Tests#
###########################
@pytest.mark.filterwarnings("ignore::DeprecationWarning")
class AutoscaleTests:
options = {"autoscale": True}
test_bug_6139 = LinprogCommonTests.test_bug_6139
test_bug_6690 = LinprogCommonTests.test_bug_6690
test_bug_7237 = LinprogCommonTests.test_bug_7237
class TestAutoscaleIP(AutoscaleTests):
method = "interior-point"
def test_bug_6139(self):
self.options['tol'] = 1e-10
return AutoscaleTests.test_bug_6139(self)
class TestAutoscaleSimplex(AutoscaleTests):
method = "simplex"
class TestAutoscaleRS(AutoscaleTests):
method = "revised simplex"
def test_nontrivial_problem_with_guess(self):
c, A_ub, b_ub, A_eq, b_eq, x_star, f_star = nontrivial_problem()
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options, x0=x_star)
_assert_success(res, desired_fun=f_star, desired_x=x_star)
assert_equal(res.nit, 0)
def test_nontrivial_problem_with_bad_guess(self):
c, A_ub, b_ub, A_eq, b_eq, x_star, f_star = nontrivial_problem()
bad_guess = [1, 2, 3, .5]
res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds,
method=self.method, options=self.options, x0=bad_guess)
assert_equal(res.status, 6)
###########################
# Redundancy Removal Tests#
###########################
@pytest.mark.filterwarnings("ignore::DeprecationWarning")
class RRTests:
method = "interior-point"
LCT = LinprogCommonTests
# these are a few of the existing tests that have redundancy
test_RR_infeasibility = LCT.test_remove_redundancy_infeasibility
test_bug_10349 = LCT.test_bug_10349
test_bug_7044 = LCT.test_bug_7044
test_NFLC = LCT.test_network_flow_limited_capacity
test_enzo_example_b = LCT.test_enzo_example_b
class TestRRSVD(RRTests):
options = {"rr_method": "SVD"}
class TestRRPivot(RRTests):
options = {"rr_method": "pivot"}
class TestRRID(RRTests):
options = {"rr_method": "ID"}
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