def poly_constrained_dual(f, gts, eqs, p=0, q=1, ell=0, X=None, slacks=False):
"""
Construct the dual SAGE-(p, q, ell) relaxation for the polynomial optimization problem
inf{ f(x) : g(x) >= 0 for g in gts,
g(x) == 0 for g in eqs,
and x in X }
where :math:`X = R^{\\texttt{f.n}}` by default.
"""
lagrangian, ineq_lag_mults, eq_lag_mults, _ = make_poly_lagrangian(f, gts, eqs, p=p, q=q)
metadata = {'lagrangian': lagrangian, 'f': f, 'gts': gts, 'eqs': eqs, 'X': X}
if ell > 0:
alpha_E_1 = hierarchy_e_k([f, f.upcast_to_polynomial(1)] + gts + eqs, k=1)
modulator = Polynomial(2 * alpha_E_1, np.ones(alpha_E_1.shape[0])) ** ell
lagrangian = lagrangian * modulator
f = f * modulator
else:
modulator = f.upcast_to_polynomial(1)
metadata['modulator'] = modulator
# In primal form, the Lagrangian is constrained to be a SAGE polynomial.
# Introduce a dual variable "v" for this constraint.
v = cl.Variable(shape=(lagrangian.m, 1), name='v')
metadata['v_poly'] = v
constraints = relative_dual_sage_poly_cone(lagrangian, v, 'Lagrangian', log_AbK=X)
for s_g, g in ineq_lag_mults:
# These generalized Lagrange multipliers "s_g" are SAGE polynomials.
# For each such multiplier, introduce an appropriate dual variable "v_g", along
# with constraints over that dual variable.
g_m = g * modulator
c_g = sym_corr.moment_reduction_array(s_g, g_m, lagrangian)
name_base = 'v_' + str(g)
if slacks:
v_g = cl.Variable(name=name_base, shape=(s_g.m, 1))
con = c_g @ v == v_g
con.name += str(g) + ' >= 0'
constraints.append(con)
else:
v_g = c_g @ v
constraints += relative_dual_sage_poly_cone(s_g, v_g,
name_base=(name_base + ' domain'), log_AbK=X)
for z_g, g in eq_lag_mults:
# These generalized Lagrange multipliers "z_g" are arbitrary polynomials.
# They dualize to homogeneous equality constraints.
g_m = g * modulator
c_g = sym_corr.moment_reduction_array(z_g, g_m, lagrangian)
con = c_g @ v == 0
con.name += str(g) + ' == 0'
constraints.append(con)
# 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 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 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 poly_solrec(prob, ineq_tol=1e-8, eq_tol=1e-6, skip_ls=False, **kwargs):
"""
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, from ``poly_constrained_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 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.
Notes
-----
This function accepts the following keyword arguments:
zero_tol : float
Used in magnitude recovery. If a component of the Lagrangian's moment vector is smaller
than this (in absolute value), pretend it's zero in the least-squares step. Defaults to 1e-20.
heuristic_signs : bool
Used in sign recovery. If True, then attempts to infer variable signs from the Lagrangian's
moment vector even when a completely consistent set of signs does not exist. Defaults to True.
all_signs : bool
Used in sign recovery. If True, then consider returning solutions which differ only by sign.
Defaults to True.
This function is implemented only for poly_constrained_relaxation (not poly_relaxation).
"""
zero_tol = kwargs['zero_tol'] if 'zero_tol' in kwargs else 1e-20
heuristic = kwargs[
'heuristic_signs'] if 'heuristic_signs' in kwargs else True
all_signs = kwargs['all_signs'] if 'all_signs' in kwargs else True
metadata = prob.metadata
f = metadata['f']
lag_gts = metadata['gts']
lag_eqs = metadata['eqs']
lagrangian = _make_dummy_lagrangian(f, lag_gts, lag_eqs)
con = prob.constraints[0]
alpha = con.alpha
dummy_modulated_lagrangian = Polynomial(
alpha, np.ones(shape=(alpha.shape[0], ))) # coefficients dont matter
modulator = metadata['modulator']
v = metadata['v_poly'].value # possible that v_sig and v are the same
if np.any(np.isnan(v)):
return []
M = moment_reduction_array(lagrangian, modulator,
dummy_modulated_lagrangian)
v_reduced = M @ v
alpha_reduced = lagrangian.alpha
mags = variable_magnitudes(con, alpha_reduced, v_reduced, zero_tol,
skip_ls)
signs = variable_sign_patterns(alpha_reduced, v_reduced, heuristic,
all_signs)
# Now we need to build the candidate solutions, and check them for feasibility.
if con.X is not None:
gts = lag_gts + [g for g in con.X.gts]
eqs = lag_eqs + [g for g in con.X.eqs]
else:
gts = lag_gts
eqs = lag_eqs
solutions = []
for mag in mags:
for sign in signs:
x = mag * sign # elementwise
if is_feasible(x, gts, eqs, ineq_tol, eq_tol):
solutions.append(x)
solutions.sort(key=lambda xi: f(xi))
return solutions