def test_unconstrained_sage_2(self):
# Background
#
# This is Example 2 from a 2018 paper by Murray, Chandrasekaran, and Wierman
# (https://arxiv.org/pdf/1810.01614.pdf).
#
# Tests
#
# (1) Check that primal / dual objective are -\infty for ell == 0.
#
# (2) Check that primal / dual objectives are close to a reference value, for ell == 1.
#
# (3) Recover a globally optimal solution at ell == 1.
#
alpha = np.array([[0, 0],
[1, 0],
[0, 1],
[1, 1],
[0.5, 1],
[1, 0.5]])
c = np.array([0, 1, 1, 1.9, -2, -2])
s = Signomial(alpha, c)
expected = [-np.inf, -0.122211863]
pd0, _ = primal_dual_vals(s, 0)
assert pd0[0] == expected[0] and pd0[1] == expected[0]
pd1, dual = primal_dual_vals(s, 1)
assert abs(pd1[0] - expected[1]) < 1e-5 and abs(pd1[1] - expected[1]) < 1e-5
solns = sig_solrec(dual)
assert s(solns[0]) < 1e-6 + dual.value
def test_conditional_constrained_sage_2(self):
# Background
#
# This is Problem 4 from a 2005 paper by Yanjun Wang and Zhian Liang.
# The optimal objective is between 11.95 and 11.96.
#
# Tests - (p, q, ell) = (0, 2, 0)
#
# (1) Verify similar primal / dual objectives.
#
# (2) Verify that primal objective is within 1 percent of a reference value.
#
# (3) Recover a solution feasible within 1e-8, within 1 percent of optimality.
#
n = 2
y = standard_sig_monomials(n)
f = 3.7 * y[0] ** 0.85 + 1.985 * y[0] + 700.3 * y[1] ** -0.75
gts = [1 - 0.7673 * y[1] ** 0.05 + 0.05 * y[0],
5 - y[0], y[0] - 0.1,
450 - y[1], y[1] - 380]
eqs = []
X = infer_domain(f, gts, eqs)
p, q, ell = 0, 2, 0
vals, dual = constrained_primal_dual_vals(f, gts, eqs, p, q, ell, X, solver='MOSEK')
assert abs(vals[0] - vals[1]) < 1e-1
assert abs(vals[0] - 11.95) / vals[0] < 1e-2
solns = sig_solrec(dual, ineq_tol=1e-8)
assert (f(solns[0]) - dual.value) / dual.value < 1e-2
def test_conditional_constrained_sage_1(self):
# Background
#
# This is Problem 1 from a 2014 paper by Xue-Ping Hou, Pei-Ping Shen, and Yong-Qiang Chen.
# It can also be found in * many * other papers.
# The optimal objective is 0.76508
#
# Tests - (p, q, ell) = (0, 1, 0).
#
# (1) Check that primal and dual objectives are close.
#
# (2) Check that primal objective is close to reference value.
#
# (3) Verify that we can recover an optimal solution.
#
n = 4
y = standard_sig_monomials(n)
f = y[2] ** 0.8 * y[3] ** 1.2
gts = [y[0] * y[3] ** -1 + y[1] ** -1 * y[3] ** -1 - 1,
- y[0] ** -2 * y[2] ** -1 - y[1] * y[2] ** -1 + 1]
gts = [-g for g in gts]
gts += [1 - y[0], y[0] - 0.1,
10 - y[1], y[1] - 5,
15 - y[2], y[2] - 8,
1 - y[3], y[3] - 0.01]
eqs = []
X = infer_domain(f, gts, eqs)
p, q, ell = 0, 1, 0
vals, dual = constrained_primal_dual_vals(f, gts, eqs, p, q, ell, X)
assert abs(vals[0] - vals[1]) < 1e-5
assert abs(vals[0] - 0.765082) < 1e-4
solns = sig_solrec(dual, ineq_tol=0)
assert f(solns[0]) < 1e-8 + dual.value
def test_conditional_sage_1(self):
# Background
#
# This is Problem 1 from MCW2019. It appears in many papers on algorithms
# for signomial progamming. All constraints are convex.
#
# Tests
#
# ell=0 and ell=3. Test for recovery of a feasible solution for
# ell=0, and test for recovery of an optimal solution for ell=3.
#
# Notes
#
# ECOS can't solve the primal when ell > 0. Only try ell=3 if MOSEK is installed.
#
y = standard_sig_monomials(3)
f = 0.5 * y[0] * y[1] ** -1 - y[0] - 5 * y[1] ** -1
gts = [100 - y[1] * y[2] ** -1 - y[1] - 0.05 * y[0] * y[2],
y[0] - 70, y[1] - 1, y[2] - 0.5,
150 - y[0], 30 - y[1], 21 - y[2]]
eqs = []
X = infer_domain(f, gts, eqs)
vals, prob = primal_dual_vals(f, 0, X)
expect = -147.85712
assert abs(vals[0] - expect) <= 1e-3
assert abs(vals[1] - expect) <= 1e-3
solutions = sig_solrec(prob)
assert len(solutions) > 0
if cl.Mosek.is_installed():
vals, prob = primal_dual_vals(f, 3, X, solver='MOSEK')
expect = -147.6666
assert abs(vals[0] - expect) <= 1e-3
assert abs(vals[1] - expect) <= 1e-3
solutions = sig_solrec(prob)
assert len(solutions) > 0
x_star = solutions[0]
gap = f(x_star) - prob.value
assert abs(gap) <= 1e-3
pass
def test_constrained_sage_1a(self):
# Tests - (p, q, ell) = (0, 1, 0)
#
# (1) Verify that primal and dual objectives are close to a reference value.
#
# (2) Recover a solution (feasible up to tol 1e-7) with at most 0.01 percent optimality gap
#
f, gs = TestSAGERelaxations._constrained_sage_1()
expected = -0.6147
actual, dual = constrained_primal_dual_vals(f, gs, [], p=0, q=1, ell=0, X=None)
assert abs(actual[0] - expected) < 1e-4 and abs(actual[1] - expected) < 1e-4
solns = sig_solrec(dual, ineq_tol=1e-7)
assert (f(solns[0]) - dual.value) / abs(dual.value) < 1e-4
def test_unconstrained_sage_1(self, presolve=False, compactdual=True, kernel_basis=False):
# Background
#
# This is Example 1 from a 2018 paper by Murray, Chandrasekaran, and Wierman
# (https://arxiv.org/pdf/1810.01614.pdf).
#
# Tests
#
# (1) Check that primal / dual objectives are close to a reference values, for ell \in {0, 1}.
#
# (2) Recover a globally optimal solution at ell == 1.
#
initial_presolve = sage_cones.SETTINGS['presolve_trivial_age_cones']
initial_compactdual = sage_cones.SETTINGS['compact_dual']
initial_kb = sage_cones.SETTINGS['kernel_basis']
cl.presolve_trivial_age_cones(presolve)
cl.compact_sage_duals(compactdual)
cl.kernel_basis_age_witnesses(kernel_basis)
alpha = np.array([[0, 0],
[1, 0],
[0, 1],
[1, 1],
[0.5, 0],
[0, 0.5]])
c = np.array([0, 3, 2, 1, -4, -2])
s = Signomial(alpha, c)
expected = [-1.83333, -1.746505595]
pd0, _ = primal_dual_vals(s, 0)
self.assertAlmostEqual(pd0[0], expected[0], 4)
self.assertAlmostEqual(pd0[1], expected[0], 4)
pd1, dual = primal_dual_vals(s, 1)
self.assertAlmostEqual(pd1[0], expected[1], 4)
self.assertAlmostEqual(pd1[1], expected[1], 4)
optsols = sig_solrec(dual)
assert (s(optsols[0]) - dual.value) < 1e-6
cl.presolve_trivial_age_cones(initial_presolve)
cl.compact_sage_duals(initial_compactdual)
cl.kernel_basis_age_witnesses(initial_kb)
def test_conditional_constrained_sage_3(self):
# Background
#
# This is a modification of Problem 10 from the 1978 paper by M. Rijckaert and X. Martens.
# The bound constraints come from a different paper (but I dont recall where...).
# We also add an additional trivially-valid constraint (simply to strengthen the Lagrange dual).
# The optimal objective is reported in the literature as approximately -83.21.
#
# Tests - (p, q, ell) = (0, 1, 1)
#
# (1) Verify similar primal / dual objectives.
#
# (2) Recover a strictly feasible solution, within 0.7 % of optimality.
#
# Notes
#
# With (p, q, ell) = (0, 0, 2) (not shown here), we get a SAGE bound of -83.253.
# If the (0,0,2) SAGE bound of -83.253 is to be believed, then the recovered solution
# actually has a relative optimality gap of only 0.003 percent.
#
n = 3
x = standard_sig_monomials(n)
f = 0.5 * x[0] * (x[1] ** -1) - x[0] - 5.0 * (x[1] ** -1)
g = 1 - 0.01 * x[1] * (x[2] ** -1) - 0.01 * x[0] - 0.0005 * x[0] * x[2]
g = 100.0 * g
gts = [g, g * (x[1] ** -2),
1e2 - x[0], 1e2 - x[1], 1e2 - x[2],
x[0] - 1, x[1] - 1, x[2] - 1]
eqs = []
X = infer_domain(f, gts, eqs)
p, q, ell = 0, 1, 1
vals, dual = constrained_primal_dual_vals(f, gts, eqs, p, q, ell, X, solver='MOSEK')
assert abs(vals[0] - vals[1]) < 1e-4
assert abs(vals[0] - (-83.3235)) < 1e-4
solns = sig_solrec(dual, ineq_tol=0)
assert (f(solns[0]) - dual.value) / abs(dual.value) < 0.007
def test_unconstrained_sage_4(self, presolve=False, compactdual=False, kernel_basis=False):
# Background
#
# This example was constructed soley as a test case for sageopt.
#
# Minimize s(x) = exp(3*x) - 4*exp(2*x) + 7*exp(x) + exp(-x), over x \in R.
#
# Tests
#
# (1) Check that primal / dual objectives are close to reference values, for ell \in {0, 1, 2}.
#
# (2) Recover a globally optimal solution from the dual relaxation, when ell == 3.
#
# Notes
#
# It may not be obvious, but the signomial "s" is actually convex!
#
initial_presolve = sage_cones.SETTINGS['presolve_trivial_age_cones']
initial_compactdual = sage_cones.SETTINGS['compact_dual']
initial_kb = sage_cones.SETTINGS['kernel_basis']
cl.presolve_trivial_age_cones(presolve)
cl.compact_sage_duals(compactdual)
cl.kernel_basis_age_witnesses(kernel_basis)
s = Signomial.from_dict({(3,): 1, (2,): -4, (1,): 7, (-1,): 1})
expected = [3.464102, 4.60250026, 4.6217973]
pds = [primal_dual_vals(s, ell) for ell in range(3)]
for ell in range(3):
assert abs(pds[ell][0][0] - expected[ell]) < 1e-5
assert abs(pds[ell][0][1] - expected[ell]) < 1e-5
dual = sig_relaxation(s, form='dual', ell=3)
dual.solve(solver='ECOS', verbose=False)
optsols = sig_solrec(dual)
assert s(optsols[0]) < 1e-6 + dual.value
cl.presolve_trivial_age_cones(initial_presolve)
cl.compact_sage_duals(initial_compactdual)
cl.kernel_basis_age_witnesses(initial_kb)