def sig_dual(f, ell=0, X=None, modulator_support=None):
f = f.without_zeros()
# Signomial definitions (for the objective).
lagrangian = f - cl.Variable(name='gamma')
if modulator_support is None:
modulator_support = lagrangian.alpha
t_mul = Signomial(modulator_support,
np.ones(modulator_support.shape[0]))**ell
metadata = {'f': f, 'lagrangian': lagrangian, 'modulator': t_mul, 'X': X}
lagrangian = lagrangian * t_mul
f_mod = f * t_mul
# C_SAGE^STAR (v must belong to the set defined by these constraints).
v = cl.Variable(shape=(lagrangian.m, 1), name='v')
con = relative_dual_sage_cone(lagrangian,
v,
name='Lagrangian SAGE dual constraint',
X=X)
constraints = [con]
# Equality constraint (for the Lagrangian to be bounded).
a = sym_corr.relative_coeff_vector(t_mul, lagrangian.alpha)
a = a.reshape(a.size, 1)
constraints.append(a.T @ v == 1)
# Objective definition and problem creation.
obj_vec = sym_corr.relative_coeff_vector(f_mod, lagrangian.alpha)
obj = obj_vec.T @ v
# Create coniclifts Problem
prob = cl.Problem(cl.MIN, obj, constraints)
prob.metadata = metadata
cl.clear_variable_indices()
return prob
def sig_constrained_primal(f, gts, eqs, p=0, q=1, ell=0, X=None):
"""
Construct the SAGE-(p, q, ell) primal problem for the signomial program
min{ f(x) : g(x) >= 0 for g in gts,
g(x) == 0 for g in eqs,
and x in X }
where X = :math:`R^{\\texttt{f.n}}` by default.
"""
lagrangian, ineq_lag_mults, _, gamma = make_sig_lagrangian(f, gts, eqs, p=p, q=q)
metadata = {'lagrangian': lagrangian, 'X': X}
if ell > 0:
alpha_E_1 = hierarchy_e_k([f, f.upcast_to_signomial(1)] + gts + eqs, k=1)
modulator = Signomial(alpha_E_1, np.ones(alpha_E_1.shape[0])) ** ell
lagrangian = lagrangian * modulator
else:
modulator = f.upcast_to_signomial(1)
metadata['modulator'] = modulator
# The Lagrangian (after possible multiplication, as above) must be a SAGE signomial.
con = primal_sage_cone(lagrangian, name='Lagrangian is SAGE', X=X)
constrs = [con]
# Lagrange multipliers (for inequality constraints) must be SAGE signomials.
expcovers = None
for i, (s_h, _) in enumerate(ineq_lag_mults):
con_name = 'SAGE multiplier for signomial inequality # ' + str(i)
con = primal_sage_cone(s_h, name=con_name, X=X, expcovers=expcovers)
expcovers = con.ech.expcovers # only * really * needed in first iteration, but keeps code flat.
constrs.append(con)
# Construct the coniclifts Problem.
prob = cl.Problem(cl.MAX, gamma, constrs)
prob.metadata = metadata
cl.clear_variable_indices()
return prob
def poly_primal(f, poly_ell=0, sigrep_ell=0, X=None):
if poly_ell == 0:
sr, _ = f.sig_rep
prob = sage_sigs.sig_primal(sr, sigrep_ell, X=X)
if AUTO_CLEAR_INDICES: # pragma:no cover
cl.clear_variable_indices()
return prob
else:
poly_modulator = f.standard_multiplier()**poly_ell
gamma = cl.Variable(shape=(), name='gamma')
lagrangian = (f - gamma) * poly_modulator
if sigrep_ell > 0:
sr, cons = lagrangian.sig_rep
sig_modulator = Signomial(sr.alpha,
np.ones(shape=(sr.m, )))**sigrep_ell
sig_under_test = sr * sig_modulator
con_name = 'Lagrangian modulated sigrep sage'
con = sage_sigs.primal_sage_cone(sig_under_test, con_name, X=X)
constraints = [con] + cons
else:
con_name = 'Lagrangian sage poly'
constraints = primal_sage_poly_cone(lagrangian,
con_name,
log_AbK=X)
obj = gamma
prob = cl.Problem(cl.MAX, obj, constraints)
if AUTO_CLEAR_INDICES: # pragma:no cover
cl.clear_variable_indices()
return prob
def _compute_sig_rep(self):
self._sig_rep = None
self._sig_rep_constrs = []
sigrep_c = np.zeros(shape=(self.m, ), dtype=object)
need_vars = []
for i, row in enumerate(self.alpha):
if np.any(row % 2 != 0):
if isinstance(self.c[i], __NUMERIC_TYPES__):
sigrep_c[i] = -abs(self.c[i])
elif self.c[i].is_constant():
sigrep_c[i] = -abs(self.c[i].value)
else:
need_vars.append(i)
else:
if isinstance(self.c[i], np.ndarray):
sigrep_c[i] = self.c[i][()]
else:
sigrep_c[i] = self.c[i]
if len(need_vars) > 0:
var_name = str(self) + ' variable sigrep coefficients'
c_hat = cl.Variable(shape=(len(need_vars), ), name=var_name)
sigrep_c[need_vars] = c_hat
self._sig_rep_constrs.append(c_hat <= self.c[need_vars])
self._sig_rep_constrs.append(c_hat <= -self.c[need_vars])
if sigrep_c.dtype == object:
sigrep_c = cl.Expression(sigrep_c)
self._sig_rep = Signomial(self.alpha, sigrep_c)
pass
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 sig_solrec(prob, ineq_tol=1e-8, eq_tol=1e-6, skip_ls=False):
"""
Recover a list of candidate solutions from a dual SAGE relaxation. Solutions are
guaranteed to be feasible up to specified tolerances, but not necessarily optimal.
Parameters
----------
prob : coniclifts.Problem
A dual-form SAGE relaxation.
ineq_tol : float
The amount by which recovered solutions can violate inequality constraints.
eq_tol : float
The amount by which recovered solutions can violate equality constraints.
skip_ls : bool
Whether or not to skip constrained least-squares solution recovery.
Returns
-------
sols : list of ndarrays
A list of feasible solutions, sorted in increasing order of objective function value.
It is possible that this list is empty, in which case no feasible solutions were recovered.
"""
con = prob.constraints[0]
if not con.name == 'Lagrangian SAGE dual constraint': # pragma: no cover
raise RuntimeError('Unexpected first constraint in dual SAGE relaxation.')
metadata = prob.metadata
f = metadata['f']
# Recover any constraints present in "prob"
lag_gts, lag_eqs = [], []
if 'gts' in metadata:
# only happens in "constrained_sage_dual".
lag_gts = metadata['gts']
lag_eqs = metadata['eqs']
lagrangian = _make_dummy_lagrangian(f, lag_gts, lag_eqs)
if con.X is None:
X_gts, X_eqs = [], []
else:
X_gts, X_eqs = con.X.gts, con.X.eqs
gts = lag_gts + X_gts
eqs = lag_eqs + X_eqs
# Search for solutions which meet the feasibility criteria
v = con.v.value
v[v < 0] = 0
if np.any(np.isnan(v)):
return None
alpha = con.alpha
dummy_modulated_lagrangian = Signomial(alpha, np.ones(shape=(alpha.shape[0],)))
alpha_reduced = lagrangian.alpha
modulator = metadata['modulator']
M = moment_reduction_array(lagrangian, modulator, dummy_modulated_lagrangian)
if skip_ls:
sols0 = []
else:
sols0 = _least_squares_solution_recovery(alpha_reduced, con, v, M, gts, eqs, ineq_tol, eq_tol)
sols1 = _dual_age_cone_solution_recovery(con, v, M, gts, eqs, ineq_tol, eq_tol)
sols = sols0 + sols1
sols.sort(key=lambda mu: f(mu))
return sols
def hierarchy_e_k(sigs, k):
alphas = [s.alpha for s in sigs]
alpha = np.vstack(alphas)
alpha = np.unique(alpha, axis=0)
c = np.ones(shape=(alpha.shape[0], ))
s = Signomial(alpha, c)
s = s**k
return s.alpha
def __pow__(self, power, modulo=None):
if self.c.dtype not in __NUMERIC_TYPES__:
raise RuntimeError(
'Cannot exponentiate polynomials with symbolic coefficients.')
temp = Signomial(self.alpha, self.c)
temp = temp**power
temp = temp.as_polynomial()
return temp
def sig_constrained_dual(f, gts, eqs, p=0, q=1, ell=0, X=None, slacks=False):
"""
Construct the SAGE-(p, q, ell) dual problem for the signomial program
min{ f(x) : g(x) >= 0 for g in gts,
g(x) == 0 for g in eqs,
and x in X }
where X = :math:`R^{\\texttt{f.n}}` by default.
"""
lagrangian, ineq_lag_mults, eq_lag_mults, _ = make_sig_lagrangian(f, gts, eqs, p=p, q=q)
metadata = {'lagrangian': lagrangian, 'f': f, 'gts': gts, 'eqs': eqs, 'level': (p, q, ell), 'X': X}
if ell > 0:
alpha_E_1 = hierarchy_e_k([f, f.upcast_to_signomial(1)] + list(gts) + list(eqs), k=1)
modulator = Signomial(alpha_E_1, np.ones(alpha_E_1.shape[0])) ** ell
lagrangian = lagrangian * modulator
f = f * modulator
else:
modulator = f.upcast_to_signomial(1)
metadata['modulator'] = modulator
# In primal form, the Lagrangian is constrained to be a SAGE signomial.
# Introduce a dual variable "v" for this constraint.
v = cl.Variable(shape=(lagrangian.m, 1), name='v')
con = relative_dual_sage_cone(lagrangian, v, name='Lagrangian SAGE dual constraint', X=X)
constraints = [con]
expcovers = None
for i, (s_h, h) in enumerate(ineq_lag_mults):
# These generalized Lagrange multipliers "s_h" are SAGE signomials.
# For each such multiplier, introduce an appropriate dual variable "v_h", along
# with constraints over that dual variable.
h_m = h * modulator
c_h = sym_corr.moment_reduction_array(s_h, h_m, lagrangian)
if slacks:
v_h = cl.Variable(name='v_' + str(h), shape=(s_h.m, 1))
constraints.append(c_h @ v == v_h)
else:
v_h = c_h @ v
con_name = 'SAGE dual for signomial inequality # ' + str(i)
con = relative_dual_sage_cone(s_h, v_h, name=con_name, X=X, expcovers=expcovers)
expcovers = con.ech.expcovers # only * really * needed in first iteration, but keeps code flat.
constraints.append(con)
for s_h, h in eq_lag_mults:
# These generalized Lagrange multipliers "s_h" are arbitrary signomials.
# They dualize to homogeneous equality constraints.
h = h * modulator
c_h = sym_corr.moment_reduction_array(s_h, h, lagrangian)
constraints.append(c_h @ v == 0)
# Equality constraint (for the Lagrangian to be bounded).
a = sym_corr.relative_coeff_vector(modulator, lagrangian.alpha)
constraints.append(a.T @ v == 1)
# Define the dual objective function.
obj_vec = sym_corr.relative_coeff_vector(f, lagrangian.alpha)
obj = obj_vec.T @ v
# Return the coniclifts Problem.
prob = cl.Problem(cl.MIN, obj, constraints)
prob.metadata = metadata
cl.clear_variable_indices()
return prob
def infer_domain(f, gts, eqs, check_feas=True):
"""
Identify a subset of the constraints in ``gts`` and ``eqs`` which can be incorporated into
conditional SAGE relaxations for polynomials. Construct a PolyDomain object from the inferred
constraints.
Parameters
----------
f : Polynomial
The objective in a desired optimization problem. This parameter is only used to determine
the dimension of the set defined by constraints in ``gts`` and ``eqs``.
gts : list of Polynomials
For every ``g in gts``, there is a desired constraint that variables ``x`` satisfy ``g(x) >= 0``.
eqs : list of Polynomials
For every ``g in eqs``, there is a desired constraint that variables ``x`` satisfy ``g(x) == 0``.
check_feas : bool
Indicates whether or not to verify that the returned PolyDomain is nonempty.
Returns
-------
X : PolyDomain or None
"""
# GP-representable inequality constraints (recast as "Signomial >= 0")
gp_gts = con_gen.valid_gp_representable_poly_inequalities(gts)
gp_gts_sigreps = [Signomial(g.alpha, g.c) for g in gp_gts]
gp_gts_sigreps = con_gen.valid_posynomial_inequalities(gp_gts_sigreps)
# ^ That second call is to convexify the signomials.
# GP-representable equality constraints (recast as "Signomial == 0")
gp_eqs = con_gen.valid_gp_representable_poly_eqs(eqs)
gp_eqs_sigreps = [Signomial(g.alpha, g.c) for g in gp_eqs]
gp_eqs_sigreps = con_gen.valid_monomial_equations(gp_eqs_sigreps)
# ^ That second call is to make sure the nonconstant term has
# a particular sign (specifically, a negative sign).
clcons = con_gen.clcons_from_standard_gprep(f.n, gp_gts_sigreps,
gp_eqs_sigreps)
if len(clcons) > 0:
polydom = PolyDomain(f.n,
logspace_cons=clcons,
gts=gp_gts,
eqs=gp_eqs,
check_feas=check_feas)
return polydom
else:
return None
def as_signomial(self):
"""
Returns
-------
f : Signomial
For every elementwise positive vector ``x``, we have ``self(x) == f(np.log(x))``.
"""
f = Signomial(self.alpha, self.c)
return f
def test_construction(self):
# data for tests
alpha = np.array([[0], [1], [2]])
c = np.array([1, -1, -2])
alpha_c = {(0, ): 1, (1, ): -1, (2, ): -2}
# Construction with two numpy arrays as arguments
s = Signomial(alpha, c)
assert s.n == 1 and s.m == 3 and s.alpha_c == alpha_c
# Construction with a vector-to-coefficient dictionary
s = Signomial.from_dict(alpha_c)
recovered_alpha_c = dict()
for i in range(s.m):
recovered_alpha_c[tuple(s.alpha[i, :])] = s.c[i]
assert s.n == 1 and s.m == 3 and alpha_c == recovered_alpha_c
def sig_primal(f, ell=0, X=None, modulator_support=None):
f = f.without_zeros()
gamma = cl.Variable(name='gamma')
lagrangian = f - gamma
if modulator_support is None:
modulator_support = lagrangian.alpha
t = Signomial(modulator_support, np.ones(modulator_support.shape[0]))
s_mod = lagrangian * (t ** ell)
con = primal_sage_cone(s_mod, name=str(s_mod), X=X)
constraints = [con]
obj = gamma.as_expr()
prob = cl.Problem(cl.MAX, obj, constraints)
cl.clear_variable_indices()
return prob
def sage_multiplier_search(f, level=1, X=None):
"""
Constructs a coniclifts maximization Problem which is feasible if ``f`` can be certified as nonnegative
over ``X``, by using an appropriate X-SAGE modulating function.
Parameters
----------
f : Signomial
We want to test if ``f`` is nonnegative over ``X``.
level : int
Controls the complexity of the X-SAGE modulating function. Must be a positive integer.
X : SigDomain
If ``X`` is None, then we test nonnegativity of ``f`` over :math:`R^{\\texttt{f.n}}`.
Returns
-------
prob : sageopt.coniclifts.Problem
Notes
-----
This function provides an alternative to moving up the reference SAGE hierarchy, for the goal of certifying
nonnegativity of a signomial ``f`` over some convex set ``X``. In general, the approach is to introduce
a signomial
``mult = Signomial(alpha_hat, c_tilde)``
where the rows of ``alpha_hat`` are all ``level``-wise sums of rows from ``f.alpha``, and ``c_tilde``
is a coniclifts Variable defining a nonzero X-SAGE function. Then we check if ``f_mod = f * mult``
is X-SAGE for any choice of ``c_tilde``.
"""
f = f.without_zeros()
constraints = []
mult_alpha = hierarchy_e_k([f, f.upcast_to_signomial(1)], k=level)
c_tilde = cl.Variable(mult_alpha.shape[0], name='c_tilde')
mult = Signomial(mult_alpha, c_tilde)
constraints.append(cl.sum(c_tilde) >= 1)
sig_under_test = mult * f
con1 = primal_sage_cone(mult, name=str(mult), X=X)
con2 = primal_sage_cone(sig_under_test, name=str(sig_under_test), X=X)
constraints.append(con1)
constraints.append(con2)
prob = cl.Problem(cl.MAX, cl.Expression([0]), constraints)
if AUTO_CLEAR_INDICES: # pragma:no cover
cl.clear_variable_indices()
return prob
def test_scalar_multiplication(self):
# data for tests
alpha0 = np.array([[0], [1], [2]])
c0 = np.array([1, 2, 3])
s0 = Signomial(alpha0, c0)
# Tests
s = 2 * s0
# noinspection PyTypeChecker
assert set(s.c) == set(2 * s0.c)
s = s0 * 2
# noinspection PyTypeChecker
assert set(s.c) == set(2 * s0.c)
s = 1 * s0
assert s.alpha_c == s0.alpha_c
s = 0 * s0
s = s.without_zeros()
assert s.m == 1 and set(s.c) == {0}
def sig_primal(f, ell=0, X=None, modulator_support=None):
f = f.without_zeros()
gamma = cl.Variable(name='gamma')
lagrangian = f - gamma
if modulator_support is None:
modulator_support = lagrangian.alpha
t = Signomial(modulator_support, np.ones(modulator_support.shape[0]))
t_mul = t**ell
s_mod = lagrangian * t_mul
con = primal_sage_cone(s_mod, name=str(s_mod), X=X)
constraints = [con]
obj = gamma.as_expr()
prob = cl.Problem(cl.MAX, obj, constraints)
prob.metadata = {
'f': f,
'lagrangian': lagrangian,
'modulator': t_mul,
'X': X
}
if AUTO_CLEAR_INDICES: # pragma:no cover
cl.clear_variable_indices()
return prob
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_unconstrained_sage_6(self):
# Background
#
# This is Example 5 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 reference values, for ell \in {0, 1}.
#
alpha = np.array([[0., 1.],
[0., 0.],
[0.52, 0.15],
[1., 0.],
[2., 2.],
[1.3, 1.38]])
c = np.array([2.55, 0.31, -1.48, 0.85, 0.65, -1.73])
s = Signomial(alpha, c)
expected = [0.00354263, 0.13793126]
pd0, _ = primal_dual_vals(s, 0)
assert abs(pd0[0] - expected[0]) < 1e-6 and abs(pd0[1] - expected[0]) < 1e-6
pd1, _ = primal_dual_vals(s, 1)
assert abs(pd1[0] - expected[1]) < 1e-6 and abs(pd1[1] - expected[1]) < 1e-6
def test_unconstrained_sage_5(self):
# Background
#
# This is Example 4 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 reference values, for ell \in {0, 1}.
#
alpha = np.array([[0., 1.],
[0.21, 0.08],
[0.16, 0.54],
[0., 0.],
[1., 0.],
[0.3, 0.58]])
c = np.array([1., -57.75, -40.37, 33.94, 67.29, 38.28])
s = Signomial(alpha, c)
expected = [-24.054866, -21.31651]
pd0, _ = primal_dual_vals(s, 0)
assert abs(pd0[0] - expected[0]) < 1e-4 and abs(pd0[1] - expected[0]) < 1e-4
pd1, _ = primal_dual_vals(s, 1)
assert abs(pd1[0] - expected[1]) < 1e-4 and abs(pd1[1] - expected[1]) < 1e-4
def make_sig_lagrangian(f, gts, eqs, p, q):
"""
Given a problem
.. math::
\\begin{align*}
\min\{ f(x) :~& g(x) \geq 0 \\text{ for } g \\in \\mathtt{gts}, \\\\
& g(x) = 0 \\text{ for } g \\in \\mathtt{eqs}, \\\\
& \\text{and } x \\in X \}
\\end{align*}
construct the q-fold constraints ``q-gts`` and ``q-eqs``, by taking all products
of ``<= q`` elements from ``gts`` and ``eqs`` respectively. Then form the Lagrangian
.. math::
L = f - \\gamma
- \sum_{g \, \\in \, \\mathtt{q-gts}} s_g \cdot g
- \sum_{g \, \\in \, \\mathtt{q-eqs}} z_g \cdot g
where :math:`\\gamma` is a coniclifts Variable of dimension 1, and the coefficients
on Signomials :math:`s_g` and :math:`z_g` are coniclifts Variables of a dimension
determined by ``p``.
Parameters
----------
f : Signomial
The objective in a desired minimization problem.
gts : list of Signomials
For every ``g in gts``, there is a desired constraint that variables ``x`` satisfy ``g(x) >= 0``.
eqs : list of Signomials
For every ``g in eqs``, there is a desired constraint that variables ``x`` satisfy ``g(x) == 0``.
p : int
Controls the complexity of ``s_g`` and ``z_g``.
q : int
The number of folds of constraints ``gts`` and ``eqs``.
Returns
-------
L : Signomial
``L.c`` is an affine expression of coniclifts Variables.
ineq_dual_sigs : a list of pairs of Signomial objects.
If the pair ``(s_g, g)`` is in this list, then ``s_g`` is a generalized Lagrange multiplier
to the constraint ``g(x) >= 0``.
eq_dual_sigs : a list of pairs of Signomial objects.
If the pair ``(z_g, g)`` is in this list, then ``z_g`` is a generalized Lagrange multiplier to the
constraint ``g(x) == 0``.
gamma : coniclifts.Variable.
In primal-form SAGE relaxations, we want to maximize ``gamma``. In dual form SAGE relaxations,
``gamma`` induces an equality constraint.
Notes
-----
The Lagrange multipliers ``s_g`` and ``z_g`` share a common matrix of exponent vectors,
which we call ``alpha_hat``.
When ``p = 0``, ``alpha_hat`` consists of a single row, of all zeros. In this case,
``s_g`` and ``z_g`` are constant Signomials, and the coefficient vectors ``s_g.c``
and ``z_g.c`` are effectively scalars. When ``p > 0``, the rows of ``alpha_hat`` are
set to all ``p``-wise sums of exponent vectors appearing in either ``f``, or some
``g in gts``, or some ``g in eqs``.
"""
folded_gt = con_gen.up_to_q_fold_cons(gts, q)
gamma = cl.Variable(name='gamma')
L = f - gamma
alpha_E_p = hierarchy_e_k([L] + list(gts) + list(eqs), k=p)
ineq_dual_sigs = []
summands = [L]
for g in folded_gt:
s_g_coeff = cl.Variable(name='s_' + str(g),
shape=(alpha_E_p.shape[0], ))
s_g = Signomial(alpha_E_p, s_g_coeff)
summands.append(-g * s_g)
ineq_dual_sigs.append((s_g, g))
eq_dual_sigs = []
folded_eq = con_gen.up_to_q_fold_cons(eqs, q)
for g in folded_eq:
z_g_coeff = cl.Variable(name='z_' + str(g),
shape=(alpha_E_p.shape[0], ))
z_g = Signomial(alpha_E_p, z_g_coeff)
summands.append(-g * z_g)
eq_dual_sigs.append((z_g, g))
L = Signomial.sum(summands)
return L, ineq_dual_sigs, eq_dual_sigs, gamma
