class ExternalPyomoModel(ExternalGreyBoxModel):
"""
This is an ExternalGreyBoxModel used to create an external model
from existing Pyomo components. Given a system of variables and
equations partitioned into "input" and "external" variables and
"residual" and "external" equations, this class computes the
residual of the "residual equations," as well as their Jacobian
and Hessian, as a function of only the inputs.
Pyomo components:
f(x, y) == 0 # "Residual equations"
g(x, y) == 0 # "External equations", dim(g) == dim(y)
Effective constraint seen by this "external model":
F(x) == f(x, y(x)) == 0
where y(x) solves g(x, y) == 0
"""
def __init__(
self,
input_vars,
external_vars,
residual_cons,
external_cons,
solver=None,
):
if solver is None:
solver = SolverFactory("ipopt")
self._solver = solver
# We only need this block to construct the NLP, which wouldn't
# be necessary if we could compute Hessians of Pyomo constraints.
self._block = create_subsystem_block(
residual_cons + external_cons,
input_vars + external_vars,
)
self._block._obj = Objective(expr=0.0)
self._nlp = PyomoNLP(self._block)
self._scc_list = list(
generate_strongly_connected_components(external_cons,
variables=external_vars))
assert len(external_vars) == len(external_cons)
self.input_vars = input_vars
self.external_vars = external_vars
self.residual_cons = residual_cons
self.external_cons = external_cons
self.residual_con_multipliers = [None for _ in residual_cons]
self.residual_scaling_factors = None
def n_inputs(self):
return len(self.input_vars)
def n_equality_constraints(self):
return len(self.residual_cons)
# I would like to try to get by without using the following "name" methods.
def input_names(self):
return ["input_%i" % i for i in range(self.n_inputs())]
def equality_constraint_names(self):
return [
"residual_%i" % i for i in range(self.n_equality_constraints())
]
def set_input_values(self, input_values):
solver = self._solver
external_cons = self.external_cons
external_vars = self.external_vars
input_vars = self.input_vars
for var, val in zip(input_vars, input_values):
var.set_value(val)
for block, inputs in self._scc_list:
if len(block.vars) == 1:
calculate_variable_from_constraint(block.vars[0],
block.cons[0])
else:
with TemporarySubsystemManager(to_fix=inputs):
solver.solve(block)
# Send updated variable values to NLP for dervative evaluation
primals = self._nlp.get_primals()
to_update = input_vars + external_vars
indices = self._nlp.get_primal_indices(to_update)
values = np.fromiter((var.value for var in to_update), float)
primals[indices] = values
self._nlp.set_primals(primals)
def set_equality_constraint_multipliers(self, eq_con_multipliers):
"""
Sets multipliers for residual equality constraints seen by the
outer solver.
"""
for i, val in enumerate(eq_con_multipliers):
self.residual_con_multipliers[i] = val
def set_external_constraint_multipliers(self, eq_con_multipliers):
eq_con_multipliers = np.array(eq_con_multipliers)
external_multipliers = self.calculate_external_constraint_multipliers(
eq_con_multipliers, )
multipliers = np.concatenate(
(eq_con_multipliers, external_multipliers))
cons = self.residual_cons + self.external_cons
n_con = len(cons)
assert n_con == self._nlp.n_constraints()
duals = np.zeros(n_con)
indices = self._nlp.get_constraint_indices(cons)
duals[indices] = multipliers
self._nlp.set_duals(duals)
def calculate_external_constraint_multipliers(self, resid_multipliers):
"""
Calculates the multipliers of the external constraints from the
multipliers of the residual constraints (which are provided by
the "outer" solver).
"""
# NOTE: This method implicitly relies on the value of inputs stored
# in the nlp. Should we also rely on the multiplier that are in
# the nlp?
# We would then need to call nlp.set_duals twice. Once with the
# residual multipliers and once with the full multipliers.
# I like the current approach better for now.
nlp = self._nlp
y = self.external_vars
f = self.residual_cons
g = self.external_cons
jfy = nlp.extract_submatrix_jacobian(y, f)
jgy = nlp.extract_submatrix_jacobian(y, g)
jgy_t = jgy.transpose()
jfy_t = jfy.transpose()
dfdg = -sps.linalg.splu(jgy_t.tocsc()).solve(jfy_t.toarray())
resid_multipliers = np.array(resid_multipliers)
external_multipliers = dfdg.dot(resid_multipliers)
return external_multipliers
def get_full_space_lagrangian_hessians(self):
"""
Calculates terms of Hessian of full-space Lagrangian due to
external and residual constraints. Note that multipliers are
set by set_equality_constraint_multipliers. These matrices
are used to calculate the Hessian of the reduced-space
Lagrangian.
"""
nlp = self._nlp
x = self.input_vars
y = self.external_vars
hlxx = nlp.extract_submatrix_hessian_lag(x, x)
hlxy = nlp.extract_submatrix_hessian_lag(x, y)
hlyy = nlp.extract_submatrix_hessian_lag(y, y)
return hlxx, hlxy, hlyy
def calculate_reduced_hessian_lagrangian(self, hlxx, hlxy, hlyy):
"""
Performs the matrix multiplications necessary to get the
reduced space Hessian-of-Lagrangian term from the full-space
terms.
"""
# Converting to dense is faster for the distillation
# example. Does this make sense?
hlxx = hlxx.toarray()
hlxy = hlxy.toarray()
hlyy = hlyy.toarray()
dydx = self.evaluate_jacobian_external_variables()
term1 = hlxx
prod = hlxy.dot(dydx)
term2 = prod + prod.transpose()
term3 = hlyy.dot(dydx).transpose().dot(dydx)
hess_lag = term1 + term2 + term3
return hess_lag
def evaluate_equality_constraints(self):
return self._nlp.extract_subvector_constraints(self.residual_cons)
def evaluate_jacobian_equality_constraints(self):
nlp = self._nlp
x = self.input_vars
y = self.external_vars
f = self.residual_cons
g = self.external_cons
jfx = nlp.extract_submatrix_jacobian(x, f)
jfy = nlp.extract_submatrix_jacobian(y, f)
jgx = nlp.extract_submatrix_jacobian(x, g)
jgy = nlp.extract_submatrix_jacobian(y, g)
nf = len(f)
nx = len(x)
n_entries = nf * nx
# TODO: Does it make sense to cast dydx to a sparse matrix?
# My intuition is that it does only if jgy is "decomposable"
# in the strongly connected component sense, which is probably
# not usually the case.
dydx = -1 * sps.linalg.splu(jgy.tocsc()).solve(jgx.toarray())
# NOTE: PyNumero block matrices require this to be a sparse matrix
# that contains coordinates for every entry that could possibly
# be nonzero. Here, this is all of the entries.
dfdx = jfx + jfy.dot(dydx)
return _dense_to_full_sparse(dfdx)
def evaluate_jacobian_external_variables(self):
nlp = self._nlp
x = self.input_vars
y = self.external_vars
g = self.external_cons
jgx = nlp.extract_submatrix_jacobian(x, g)
jgy = nlp.extract_submatrix_jacobian(y, g)
jgy_csc = jgy.tocsc()
dydx = -1 * sps.linalg.splu(jgy_csc).solve(jgx.toarray())
return dydx
def evaluate_hessian_external_variables(self):
nlp = self._nlp
x = self.input_vars
y = self.external_vars
g = self.external_cons
jgx = nlp.extract_submatrix_jacobian(x, g)
jgy = nlp.extract_submatrix_jacobian(y, g)
jgy_csc = jgy.tocsc()
jgy_fact = sps.linalg.splu(jgy_csc)
dydx = -1 * jgy_fact.solve(jgx.toarray())
ny = len(y)
nx = len(x)
hgxx = np.array([
get_hessian_of_constraint(con, x, nlp=nlp).toarray() for con in g
])
hgxy = np.array([
get_hessian_of_constraint(con, x, y, nlp=nlp).toarray()
for con in g
])
hgyy = np.array([
get_hessian_of_constraint(con, y, nlp=nlp).toarray() for con in g
])
# This term is sparse, but we do not exploit it.
term1 = hgxx
# This is what we want.
# prod[i,j,k] = sum(hgxy[i,:,j] * dydx[:,k])
prod = hgxy.dot(dydx)
# Swap the second and third axes of the tensor
term2 = prod + prod.transpose((0, 2, 1))
# The term2 tensor could have some sparsity worth exploiting.
# matrix.dot(tensor) is not what we want, so we reverse the order of the
# product. Exploit symmetry of hgyy to only perform one transpose.
term3 = hgyy.dot(dydx).transpose((0, 2, 1)).dot(dydx)
rhs = term1 + term2 + term3
rhs.shape = (ny, nx * nx)
sol = jgy_fact.solve(rhs)
sol.shape = (ny, nx, nx)
d2ydx2 = -sol
return d2ydx2
def evaluate_hessians_of_residuals(self):
"""
This method computes the Hessian matrix of each equality
constraint individually, rather than the sum of Hessians
times multipliers.
"""
nlp = self._nlp
x = self.input_vars
y = self.external_vars
f = self.residual_cons
g = self.external_cons
jfx = nlp.extract_submatrix_jacobian(x, f)
jfy = nlp.extract_submatrix_jacobian(y, f)
dydx = self.evaluate_jacobian_external_variables()
ny = len(y)
nf = len(f)
nx = len(x)
hfxx = np.array([
get_hessian_of_constraint(con, x, nlp=nlp).toarray() for con in f
])
hfxy = np.array([
get_hessian_of_constraint(con, x, y, nlp=nlp).toarray()
for con in f
])
hfyy = np.array([
get_hessian_of_constraint(con, y, nlp=nlp).toarray() for con in f
])
d2ydx2 = self.evaluate_hessian_external_variables()
term1 = hfxx
prod = hfxy.dot(dydx)
term2 = prod + prod.transpose((0, 2, 1))
term3 = hfyy.dot(dydx).transpose((0, 2, 1)).dot(dydx)
d2ydx2.shape = (ny, nx * nx)
term4 = jfy.dot(d2ydx2)
term4.shape = (nf, nx, nx)
d2fdx2 = term1 + term2 + term3 + term4
return d2fdx2
def evaluate_hessian_equality_constraints(self):
"""
This method actually evaluates the sum of Hessians times
multipliers, i.e. the term in the Hessian of the Lagrangian
due to these equality constraints.
"""
# External multipliers must be calculated after both primals and duals
# are set, and are only necessary for this Hessian calculation.
# We know this Hessian calculation wants to use the most recently
# set primals and duals, so we can safely calculate external
# multipliers here.
eq_con_multipliers = self.residual_con_multipliers
self.set_external_constraint_multipliers(eq_con_multipliers)
# These are full-space Hessian-of-Lagrangian terms
hlxx, hlxy, hlyy = self.get_full_space_lagrangian_hessians()
# These terms can be used to calculate the corresponding
# Hessian-of-Lagrangian term in the full space.
hess_lag = self.calculate_reduced_hessian_lagrangian(hlxx, hlxy, hlyy)
sparse = _dense_to_full_sparse(hess_lag)
return sps.tril(sparse)
def set_equality_constraint_scaling_factors(self, scaling_factors):
"""
Set scaling factors for the equality constraints that are exposed
to a solver. These are the "residual equations" in this class.
"""
self.residual_scaling_factors = np.array(scaling_factors)
def get_equality_constraint_scaling_factors(self):
"""
Get scaling factors for the equality constraints that are exposed
to a solver. These are the "residual equations" in this class.
"""
return self.residual_scaling_factors
def test_indices_methods(self):
nlp = PyomoNLP(self.pm)
# get_pyomo_variables
variables = nlp.get_pyomo_variables()
expected_ids = [id(self.pm.x[i]) for i in range(1, 10)]
ids = [id(variables[i]) for i in range(9)]
self.assertTrue(expected_ids == ids)
variable_names = nlp.variable_names()
expected_names = [self.pm.x[i].getname() for i in range(1, 10)]
self.assertTrue(variable_names == expected_names)
# get_pyomo_constraints
constraints = nlp.get_pyomo_constraints()
expected_ids = [id(self.pm.c[i]) for i in range(1, 10)]
ids = [id(constraints[i]) for i in range(9)]
self.assertTrue(expected_ids == ids)
constraint_names = nlp.constraint_names()
expected_names = [c.getname() for c in nlp.get_pyomo_constraints()]
self.assertTrue(constraint_names == expected_names)
# get_pyomo_equality_constraints
eq_constraints = nlp.get_pyomo_equality_constraints()
# 2 and 6 are the equality constraints
eq_indices = [2, 6] # "indices" here is a bit overloaded
expected_eq_ids = [id(self.pm.c[i]) for i in eq_indices]
eq_ids = [id(con) for con in eq_constraints]
self.assertEqual(eq_ids, expected_eq_ids)
eq_constraint_names = nlp.equality_constraint_names()
expected_eq_names = [
c.getname(fully_qualified=True)
for c in nlp.get_pyomo_equality_constraints()
]
self.assertEqual(eq_constraint_names, expected_eq_names)
# get_pyomo_inequality_constraints
ineq_constraints = nlp.get_pyomo_inequality_constraints()
# 1, 3, 4, 5, 7, 8, and 9 are the inequality constraints
ineq_indices = [1, 3, 4, 5, 7, 8, 9]
expected_ineq_ids = [id(self.pm.c[i]) for i in ineq_indices]
ineq_ids = [id(con) for con in ineq_constraints]
self.assertEqual(eq_ids, expected_eq_ids)
# get_primal_indices
expected_primal_indices = [i for i in range(9)]
self.assertTrue(
expected_primal_indices == nlp.get_primal_indices([self.pm.x]))
expected_primal_indices = [0, 3, 8, 4]
variables = [self.pm.x[1], self.pm.x[4], self.pm.x[9], self.pm.x[5]]
self.assertTrue(
expected_primal_indices == nlp.get_primal_indices(variables))
# get_constraint_indices
expected_constraint_indices = [i for i in range(9)]
self.assertTrue(expected_constraint_indices ==
nlp.get_constraint_indices([self.pm.c]))
expected_constraint_indices = [0, 3, 8, 4]
constraints = [self.pm.c[1], self.pm.c[4], self.pm.c[9], self.pm.c[5]]
self.assertTrue(expected_constraint_indices ==
nlp.get_constraint_indices(constraints))
# get_equality_constraint_indices
pyomo_eq_indices = [2, 6]
with self.assertRaises(KeyError):
# At least one data object in container is not an equality
nlp.get_equality_constraint_indices([self.pm.c])
eq_constraints = [self.pm.c[i] for i in pyomo_eq_indices]
expected_eq_indices = [0, 1]
# ^indices in the list of equality constraints
eq_constraint_indices = nlp.get_equality_constraint_indices(
eq_constraints)
self.assertEqual(expected_eq_indices, eq_constraint_indices)
# get_inequality_constraint_indices
pyomo_ineq_indices = [1, 3, 4, 5, 7, 9]
with self.assertRaises(KeyError):
# At least one data object in container is not an equality
nlp.get_inequality_constraint_indices([self.pm.c])
ineq_constraints = [self.pm.c[i] for i in pyomo_ineq_indices]
expected_ineq_indices = [0, 1, 2, 3, 4, 6]
# ^indices in the list of equality constraints; didn't include 8
ineq_constraint_indices = nlp.get_inequality_constraint_indices(
ineq_constraints)
self.assertEqual(expected_ineq_indices, ineq_constraint_indices)
# extract_subvector_grad_objective
expected_gradient = np.asarray(
[2 * sum((i + 1) * (j + 1) for j in range(9)) for i in range(9)],
dtype=np.float64)
grad_obj = nlp.extract_subvector_grad_objective([self.pm.x])
self.assertTrue(np.array_equal(expected_gradient, grad_obj))
expected_gradient = np.asarray([
2 * sum((i + 1) * (j + 1) for j in range(9)) for i in [0, 3, 8, 4]
],
dtype=np.float64)
variables = [self.pm.x[1], self.pm.x[4], self.pm.x[9], self.pm.x[5]]
grad_obj = nlp.extract_subvector_grad_objective(variables)
self.assertTrue(np.array_equal(expected_gradient, grad_obj))
# extract_subvector_constraints
expected_con = np.asarray(
[45, 88, 3 * 45, 4 * 45, 5 * 45, 276, 7 * 45, 8 * 45, 9 * 45],
dtype=np.float64)
con = nlp.extract_subvector_constraints([self.pm.c])
self.assertTrue(np.array_equal(expected_con, con))
expected_con = np.asarray([45, 4 * 45, 9 * 45, 5 * 45],
dtype=np.float64)
constraints = [self.pm.c[1], self.pm.c[4], self.pm.c[9], self.pm.c[5]]
con = nlp.extract_subvector_constraints(constraints)
self.assertTrue(np.array_equal(expected_con, con))
# extract_submatrix_jacobian
expected_jac = [[(i) * (j) for j in range(1, 10)]
for i in range(1, 10)]
expected_jac = np.asarray(expected_jac, dtype=np.float64)
jac = nlp.extract_submatrix_jacobian(pyomo_variables=[self.pm.x],
pyomo_constraints=[self.pm.c])
dense_jac = jac.todense()
self.assertTrue(np.array_equal(dense_jac, expected_jac))
expected_jac = [[(i) * (j) for j in [1, 4, 9, 5]] for i in [2, 6, 4]]
expected_jac = np.asarray(expected_jac, dtype=np.float64)
variables = [self.pm.x[1], self.pm.x[4], self.pm.x[9], self.pm.x[5]]
constraints = [self.pm.c[2], self.pm.c[6], self.pm.c[4]]
jac = nlp.extract_submatrix_jacobian(pyomo_variables=variables,
pyomo_constraints=constraints)
dense_jac = jac.todense()
self.assertTrue(np.array_equal(dense_jac, expected_jac))
# extract_submatrix_hessian_lag
expected_hess = [[2.0 * i * j for j in range(1, 10)]
for i in range(1, 10)]
expected_hess = np.asarray(expected_hess, dtype=np.float64)
hess = nlp.extract_submatrix_hessian_lag(
pyomo_variables_rows=[self.pm.x], pyomo_variables_cols=[self.pm.x])
dense_hess = hess.todense()
self.assertTrue(np.array_equal(dense_hess, expected_hess))
expected_hess = [[2.0 * i * j for j in [1, 4, 9, 5]]
for i in [1, 4, 9, 5]]
expected_hess = np.asarray(expected_hess, dtype=np.float64)
variables = [self.pm.x[1], self.pm.x[4], self.pm.x[9], self.pm.x[5]]
hess = nlp.extract_submatrix_hessian_lag(
pyomo_variables_rows=variables, pyomo_variables_cols=variables)
dense_hess = hess.todense()
self.assertTrue(np.array_equal(dense_hess, expected_hess))
class ExternalPyomoModel(ExternalGreyBoxModel):
"""
This is an ExternalGreyBoxModel used to create an exteral model
from existing Pyomo components. Given a system of variables and
equations partitioned into "input" and "external" variables and
"residual" and "external" equations, this class computes the
residual of the "residual equations," as well as their Jacobian
and Hessian, as a function of only the inputs.
Pyomo components:
f(x, y) == 0 # "Residual equations"
g(x, y) == 0 # "External equations", dim(g) == dim(y)
Effective constraint seen by this "external model":
F(x) == f(x, y(x)) == 0
where y(x) solves g(x, y) == 0
"""
def __init__(
self,
input_vars,
external_vars,
residual_cons,
external_cons,
solver=None,
):
if solver is None:
solver = SolverFactory("ipopt")
self._solver = solver
# We only need this block to construct the NLP, which wouldn't
# be necessary if we could compute Hessians of Pyomo constraints.
self._block = create_subsystem_block(
residual_cons + external_cons,
input_vars + external_vars,
)
self._block._obj = Objective(expr=0.0)
self._nlp = PyomoNLP(self._block)
assert len(external_vars) == len(external_cons)
self.input_vars = input_vars
self.external_vars = external_vars
self.residual_cons = residual_cons
self.external_cons = external_cons
self.residual_con_multipliers = [None for _ in residual_cons]
def n_inputs(self):
return len(self.input_vars)
def n_equality_constraints(self):
return len(self.residual_cons)
# I would like to try to get by without using the following "name" methods.
def input_names(self):
return ["input_%i" % i for i in range(self.n_inputs())]
def equality_constraint_names(self):
return [
"residual_%i" % i for i in range(self.n_equality_constraints())
]
def set_input_values(self, input_values):
solver = self._solver
external_cons = self.external_cons
external_vars = self.external_vars
input_vars = self.input_vars
for var, val in zip(input_vars, input_values):
var.set_value(val)
_temp = create_subsystem_block(external_cons, variables=external_vars)
possible_input_vars = ComponentSet(input_vars)
#for var in _temp.input_vars.values():
# # TODO: Is this check necessary?
# assert var in possible_input_vars
with TemporarySubsystemManager(to_fix=list(_temp.input_vars.values())):
solver.solve(_temp)
# Should we create the NLP from the original block or the temp block?
# Need to create it from the original block because temp block won't
# have residual constraints, whose derivatives are necessary.
self._nlp = PyomoNLP(self._block)
def set_equality_constraint_multipliers(self, eq_con_multipliers):
for i, val in enumerate(eq_con_multipliers):
self.residual_con_multipliers[i] = val
def evaluate_equality_constraints(self):
return self._nlp.extract_subvector_constraints(self.residual_cons)
def evaluate_jacobian_equality_constraints(self):
nlp = self._nlp
x = self.input_vars
y = self.external_vars
f = self.residual_cons
g = self.external_cons
jfx = nlp.extract_submatrix_jacobian(x, f)
jfy = nlp.extract_submatrix_jacobian(y, f)
jgx = nlp.extract_submatrix_jacobian(x, g)
jgy = nlp.extract_submatrix_jacobian(y, g)
nf = len(f)
nx = len(x)
n_entries = nf * nx
# TODO: Does it make sense to cast dydx to a sparse matrix?
# My intuition is that it does only if jgy is "decomposable"
# in the strongly connected component sense, which is probably
# not usually the case.
dydx = -1 * sps.linalg.splu(jgy.tocsc()).solve(jgx.toarray())
# NOTE: PyNumero block matrices require this to be a sparse matrix
# that contains coordinates for every entry that could possibly
# be nonzero. Here, this is all of the entries.
dfdx = jfx + jfy.dot(dydx)
return _dense_to_full_sparse(dfdx)
def evaluate_jacobian_external_variables(self):
nlp = self._nlp
x = self.input_vars
y = self.external_vars
g = self.external_cons
jgx = nlp.extract_submatrix_jacobian(x, g)
jgy = nlp.extract_submatrix_jacobian(y, g)
jgy_csc = jgy.tocsc()
dydx = -1 * sps.linalg.splu(jgy_csc).solve(jgx.toarray())
return dydx
def evaluate_hessian_external_variables(self):
nlp = self._nlp
x = self.input_vars
y = self.external_vars
g = self.external_cons
jgx = nlp.extract_submatrix_jacobian(x, g)
jgy = nlp.extract_submatrix_jacobian(y, g)
jgy_csc = jgy.tocsc()
jgy_fact = sps.linalg.splu(jgy_csc)
dydx = -1 * jgy_fact.solve(jgx.toarray())
ny = len(y)
nx = len(x)
hgxx = np.array([
get_hessian_of_constraint(con, x, nlp=nlp).toarray() for con in g
])
hgxy = np.array([
get_hessian_of_constraint(con, x, y, nlp=nlp).toarray()
for con in g
])
hgyy = np.array([
get_hessian_of_constraint(con, y, nlp=nlp).toarray() for con in g
])
# This term is sparse, but we do not exploit it.
term1 = hgxx
# This is what we want.
# prod[i,j,k] = sum(hgxy[i,:,j] * dydx[:,k])
prod = hgxy.dot(dydx)
# Swap the second and third axes of the tensor
term2 = prod + prod.transpose((0, 2, 1))
# The term2 tensor could have some sparsity worth exploiting.
# matrix.dot(tensor) is not what we want, so we reverse the order of the
# product. Exploit symmetry of hgyy to only perform one transpose.
term3 = hgyy.dot(dydx).transpose((0, 2, 1)).dot(dydx)
rhs = term1 + term2 + term3
rhs.shape = (ny, nx * nx)
sol = jgy_fact.solve(rhs)
sol.shape = (ny, nx, nx)
d2ydx2 = -sol
return d2ydx2
def evaluate_hessians_of_residuals(self):
"""
This method computes the Hessian matrix of each equality
constraint individually, rather than the sum of Hessians
times multipliers.
"""
nlp = self._nlp
x = self.input_vars
y = self.external_vars
f = self.residual_cons
g = self.external_cons
jfx = nlp.extract_submatrix_jacobian(x, f)
jfy = nlp.extract_submatrix_jacobian(y, f)
dydx = self.evaluate_jacobian_external_variables()
ny = len(y)
nf = len(f)
nx = len(x)
hfxx = np.array([
get_hessian_of_constraint(con, x, nlp=nlp).toarray() for con in f
])
hfxy = np.array([
get_hessian_of_constraint(con, x, y, nlp=nlp).toarray()
for con in f
])
hfyy = np.array([
get_hessian_of_constraint(con, y, nlp=nlp).toarray() for con in f
])
d2ydx2 = self.evaluate_hessian_external_variables()
term1 = hfxx
prod = hfxy.dot(dydx)
term2 = prod + prod.transpose((0, 2, 1))
term3 = hfyy.dot(dydx).transpose((0, 2, 1)).dot(dydx)
d2ydx2.shape = (ny, nx * nx)
term4 = jfy.dot(d2ydx2)
term4.shape = (nf, nx, nx)
d2fdx2 = term1 + term2 + term3 + term4
return d2fdx2
def evaluate_hessian_equality_constraints(self):
"""
This method actually evaluates the sum of Hessians times
multipliers, i.e. the term in the Hessian of the Lagrangian
due to these equality constraints.
"""
d2fdx2 = self.evaluate_hessians_of_residuals()
multipliers = self.residual_con_multipliers
sum_ = sum(mult * matrix for mult, matrix in zip(multipliers, d2fdx2))
# Return a sparse matrix with every entry accounted for because it
# is difficult to determine rigorously which coordinates
# _could possibly_ be nonzero.
sparse = _dense_to_full_sparse(sum_)
return sps.tril(sparse)
def test_indices_methods(self):
nlp = PyomoNLP(self.pm)
# get_pyomo_variables
variables = nlp.get_pyomo_variables()
expected_ids = [id(self.pm.x[i]) for i in range(1, 10)]
ids = [id(variables[i]) for i in range(9)]
self.assertTrue(expected_ids == ids)
variable_names = nlp.variable_names()
expected_names = [self.pm.x[i].getname() for i in range(1, 10)]
self.assertTrue(variable_names == expected_names)
# get_pyomo_constraints
constraints = nlp.get_pyomo_constraints()
expected_ids = [id(self.pm.c[i]) for i in range(1, 10)]
ids = [id(constraints[i]) for i in range(9)]
self.assertTrue(expected_ids == ids)
constraint_names = nlp.constraint_names()
expected_names = [c.getname() for c in nlp.get_pyomo_constraints()]
self.assertTrue(constraint_names == expected_names)
# get_primal_indices
expected_primal_indices = [i for i in range(9)]
self.assertTrue(
expected_primal_indices == nlp.get_primal_indices([self.pm.x]))
expected_primal_indices = [0, 3, 8, 4]
variables = [self.pm.x[1], self.pm.x[4], self.pm.x[9], self.pm.x[5]]
self.assertTrue(
expected_primal_indices == nlp.get_primal_indices(variables))
# get_constraint_indices
expected_constraint_indices = [i for i in range(9)]
self.assertTrue(expected_constraint_indices ==
nlp.get_constraint_indices([self.pm.c]))
expected_constraint_indices = [0, 3, 8, 4]
constraints = [self.pm.c[1], self.pm.c[4], self.pm.c[9], self.pm.c[5]]
self.assertTrue(expected_constraint_indices ==
nlp.get_constraint_indices(constraints))
# extract_subvector_grad_objective
expected_gradient = np.asarray(
[2 * sum((i + 1) * (j + 1) for j in range(9)) for i in range(9)],
dtype=np.float64)
grad_obj = nlp.extract_subvector_grad_objective([self.pm.x])
self.assertTrue(np.array_equal(expected_gradient, grad_obj))
expected_gradient = np.asarray([
2 * sum((i + 1) * (j + 1) for j in range(9)) for i in [0, 3, 8, 4]
],
dtype=np.float64)
variables = [self.pm.x[1], self.pm.x[4], self.pm.x[9], self.pm.x[5]]
grad_obj = nlp.extract_subvector_grad_objective(variables)
self.assertTrue(np.array_equal(expected_gradient, grad_obj))
# extract_subvector_constraints
expected_con = np.asarray(
[45, 88, 3 * 45, 4 * 45, 5 * 45, 276, 7 * 45, 8 * 45, 9 * 45],
dtype=np.float64)
con = nlp.extract_subvector_constraints([self.pm.c])
self.assertTrue(np.array_equal(expected_con, con))
expected_con = np.asarray([45, 4 * 45, 9 * 45, 5 * 45],
dtype=np.float64)
constraints = [self.pm.c[1], self.pm.c[4], self.pm.c[9], self.pm.c[5]]
con = nlp.extract_subvector_constraints(constraints)
self.assertTrue(np.array_equal(expected_con, con))
# extract_submatrix_jacobian
expected_jac = [[(i) * (j) for j in range(1, 10)]
for i in range(1, 10)]
expected_jac = np.asarray(expected_jac, dtype=np.float64)
jac = nlp.extract_submatrix_jacobian(pyomo_variables=[self.pm.x],
pyomo_constraints=[self.pm.c])
dense_jac = jac.todense()
self.assertTrue(np.array_equal(dense_jac, expected_jac))
expected_jac = [[(i) * (j) for j in [1, 4, 9, 5]] for i in [2, 6, 4]]
expected_jac = np.asarray(expected_jac, dtype=np.float64)
variables = [self.pm.x[1], self.pm.x[4], self.pm.x[9], self.pm.x[5]]
constraints = [self.pm.c[2], self.pm.c[6], self.pm.c[4]]
jac = nlp.extract_submatrix_jacobian(pyomo_variables=variables,
pyomo_constraints=constraints)
dense_jac = jac.todense()
self.assertTrue(np.array_equal(dense_jac, expected_jac))
# extract_submatrix_hessian_lag
expected_hess = [[2.0 * i * j for j in range(1, 10)]
for i in range(1, 10)]
expected_hess = np.asarray(expected_hess, dtype=np.float64)
hess = nlp.extract_submatrix_hessian_lag(
pyomo_variables_rows=[self.pm.x], pyomo_variables_cols=[self.pm.x])
dense_hess = hess.todense()
self.assertTrue(np.array_equal(dense_hess, expected_hess))
expected_hess = [[2.0 * i * j for j in [1, 4, 9, 5]]
for i in [1, 4, 9, 5]]
expected_hess = np.asarray(expected_hess, dtype=np.float64)
variables = [self.pm.x[1], self.pm.x[4], self.pm.x[9], self.pm.x[5]]
hess = nlp.extract_submatrix_hessian_lag(
pyomo_variables_rows=variables, pyomo_variables_cols=variables)
dense_hess = hess.todense()
self.assertTrue(np.array_equal(dense_hess, expected_hess))
class ExternalPyomoModel(ExternalGreyBoxModel):
"""
This is an ExternalGreyBoxModel used to create an external model
from existing Pyomo components. Given a system of variables and
equations partitioned into "input" and "external" variables and
"residual" and "external" equations, this class computes the
residual of the "residual equations," as well as their Jacobian
and Hessian, as a function of only the inputs.
Pyomo components:
f(x, y) == 0 # "Residual equations"
g(x, y) == 0 # "External equations", dim(g) == dim(y)
Effective constraint seen by this "external model":
F(x) == f(x, y(x)) == 0
where y(x) solves g(x, y) == 0
"""
def __init__(
self,
input_vars,
external_vars,
residual_cons,
external_cons,
use_cyipopt=None,
solver=None,
):
"""
Arguments:
----------
input_vars: list
List of variables sent to this system by the outer solver
external_vars: list
List of variables that are solved for internally by this system
residual_cons: list
List of equality constraints whose residuals are exposed to
the outer solver
external_cons: list
List of equality constraints used to solve for the external
variables
use_cyipopt: bool
Whether to use CyIpopt to solve strongly connected components of
the implicit function that have dimension greater than one.
solver: Pyomo solver object
Used to solve strongly connected components of the implicit function
that have dimension greater than one. Only used if use_cyipopt
is False.
"""
if use_cyipopt is None:
use_cyipopt = cyipopt_available
if use_cyipopt and not cyipopt_available:
raise RuntimeError(
"Constructing an ExternalPyomoModel with CyIpopt unavailable. "
"Please set the use_cyipopt argument to False.")
if solver is not None and use_cyipopt:
raise RuntimeError(
"Constructing an ExternalPyomoModel with a solver specified "
"and use_cyipopt set to True. Please set use_cyipopt to False "
"to use the desired solver.")
elif solver is None and not use_cyipopt:
solver = SolverFactory("ipopt")
# If use_cyipopt is True, this solver is None and will not be used.
self._solver = solver
self._use_cyipopt = use_cyipopt
# We only need this block to construct the NLP, which wouldn't
# be necessary if we could compute Hessians of Pyomo constraints.
self._block = create_subsystem_block(
residual_cons + external_cons,
input_vars + external_vars,
)
self._block._obj = Objective(expr=0.0)
self._nlp = PyomoNLP(self._block)
self._scc_list = list(
generate_strongly_connected_components(external_cons,
variables=external_vars))
if use_cyipopt:
# Using CyIpopt allows us to solve inner problems without
# costly rewriting of the nl file. It requires quite a bit
# of preprocessing, however, to construct the ProjectedNLP
# for each block of the decomposition.
# Get "vector-valued" SCCs, those of dimension > 0.
# We will solve these with a direct IPOPT interface, which requires
# some preprocessing.
self._vector_scc_list = [(scc, inputs)
for scc, inputs in self._scc_list
if len(scc.vars) > 1]
# Need a dummy objective to create an NLP
for scc, inputs in self._vector_scc_list:
scc._obj = Objective(expr=0.0)
# I need scaling_factor so Pyomo NLPs I create from these blocks
# don't break when ProjectedNLP calls get_primals_scaling
scc.scaling_factor = Suffix(direction=Suffix.EXPORT)
# HACK: scaling_factor just needs to be nonempty.
scc.scaling_factor[scc._obj] = 1.0
# These are the "original NLPs" that will be projected
self._vector_scc_nlps = [
PyomoNLP(scc) for scc, inputs in self._vector_scc_list
]
self._vector_scc_var_names = [[
var.name for var in scc.vars.values()
] for scc, inputs in self._vector_scc_list]
self._vector_proj_nlps = [
ProjectedNLP(nlp, names) for nlp, names in zip(
self._vector_scc_nlps, self._vector_scc_var_names)
]
# We will solve the ProjectedNLPs rather than the original NLPs
self._cyipopt_nlps = [
CyIpoptNLP(nlp) for nlp in self._vector_proj_nlps
]
self._cyipopt_solvers = [
CyIpoptSolver(nlp) for nlp in self._cyipopt_nlps
]
self._vector_scc_input_coords = [
nlp.get_primal_indices(inputs) for nlp, (scc, inputs) in zip(
self._vector_scc_nlps, self._vector_scc_list)
]
assert len(external_vars) == len(external_cons)
self.input_vars = input_vars
self.external_vars = external_vars
self.residual_cons = residual_cons
self.external_cons = external_cons
self.residual_con_multipliers = [None for _ in residual_cons]
self.residual_scaling_factors = None
def n_inputs(self):
return len(self.input_vars)
def n_equality_constraints(self):
return len(self.residual_cons)
# I would like to try to get by without using the following "name" methods.
def input_names(self):
return ["input_%i" % i for i in range(self.n_inputs())]
def equality_constraint_names(self):
return [
"residual_%i" % i for i in range(self.n_equality_constraints())
]
def set_input_values(self, input_values):
solver = self._solver
external_cons = self.external_cons
external_vars = self.external_vars
input_vars = self.input_vars
for var, val in zip(input_vars, input_values):
var.set_value(val, skip_validation=True)
vector_scc_idx = 0
for block, inputs in self._scc_list:
if len(block.vars) == 1:
calculate_variable_from_constraint(block.vars[0],
block.cons[0])
else:
if self._use_cyipopt:
# Transfer variable values into the projected NLP, solve,
# and extract values.
nlp = self._vector_scc_nlps[vector_scc_idx]
proj_nlp = self._vector_proj_nlps[vector_scc_idx]
input_coords = self._vector_scc_input_coords[
vector_scc_idx]
cyipopt = self._cyipopt_solvers[vector_scc_idx]
_, local_inputs = self._vector_scc_list[vector_scc_idx]
primals = nlp.get_primals()
variables = nlp.get_pyomo_variables()
# Set values and bounds from inputs to the SCC.
# This works because values have been set in the original
# pyomo model, either by a previous SCC solve, or from the
# "global inputs"
for i, var in zip(input_coords, local_inputs):
# Set primals (inputs) in the original NLP
primals[i] = var.value
# This affects future evaluations in the ProjectedNLP
nlp.set_primals(primals)
x0 = proj_nlp.get_primals()
sol, _ = cyipopt.solve(x0=x0)
# Set primals from solution in projected NLP. This updates
# values in the original NLP
proj_nlp.set_primals(sol)
# I really only need to set new primals for the variables in
# the ProjectedNLP. However, I can only get a list of variables
# from the original Pyomo NLP, so here some of the values I'm
# setting are redundant.
new_primals = nlp.get_primals()
assert len(new_primals) == len(variables)
for var, val in zip(variables, new_primals):
var.set_value(val, skip_validation=True)
else:
# Use a Pyomo solver to solve this strongly connected
# component.
with TemporarySubsystemManager(to_fix=inputs):
solver.solve(block)
vector_scc_idx += 1
# Send updated variable values to NLP for dervative evaluation
primals = self._nlp.get_primals()
to_update = input_vars + external_vars
indices = self._nlp.get_primal_indices(to_update)
values = np.fromiter((var.value for var in to_update), float)
primals[indices] = values
self._nlp.set_primals(primals)
def set_equality_constraint_multipliers(self, eq_con_multipliers):
"""
Sets multipliers for residual equality constraints seen by the
outer solver.
"""
for i, val in enumerate(eq_con_multipliers):
self.residual_con_multipliers[i] = val
def set_external_constraint_multipliers(self, eq_con_multipliers):
eq_con_multipliers = np.array(eq_con_multipliers)
external_multipliers = self.calculate_external_constraint_multipliers(
eq_con_multipliers, )
multipliers = np.concatenate(
(eq_con_multipliers, external_multipliers))
cons = self.residual_cons + self.external_cons
n_con = len(cons)
assert n_con == self._nlp.n_constraints()
duals = np.zeros(n_con)
indices = self._nlp.get_constraint_indices(cons)
duals[indices] = multipliers
self._nlp.set_duals(duals)
def calculate_external_constraint_multipliers(self, resid_multipliers):
"""
Calculates the multipliers of the external constraints from the
multipliers of the residual constraints (which are provided by
the "outer" solver).
"""
# NOTE: This method implicitly relies on the value of inputs stored
# in the nlp. Should we also rely on the multiplier that are in
# the nlp?
# We would then need to call nlp.set_duals twice. Once with the
# residual multipliers and once with the full multipliers.
# I like the current approach better for now.
nlp = self._nlp
y = self.external_vars
f = self.residual_cons
g = self.external_cons
jfy = nlp.extract_submatrix_jacobian(y, f)
jgy = nlp.extract_submatrix_jacobian(y, g)
jgy_t = jgy.transpose()
jfy_t = jfy.transpose()
dfdg = -sps.linalg.splu(jgy_t.tocsc()).solve(jfy_t.toarray())
resid_multipliers = np.array(resid_multipliers)
external_multipliers = dfdg.dot(resid_multipliers)
return external_multipliers
def get_full_space_lagrangian_hessians(self):
"""
Calculates terms of Hessian of full-space Lagrangian due to
external and residual constraints. Note that multipliers are
set by set_equality_constraint_multipliers. These matrices
are used to calculate the Hessian of the reduced-space
Lagrangian.
"""
nlp = self._nlp
x = self.input_vars
y = self.external_vars
hlxx = nlp.extract_submatrix_hessian_lag(x, x)
hlxy = nlp.extract_submatrix_hessian_lag(x, y)
hlyy = nlp.extract_submatrix_hessian_lag(y, y)
return hlxx, hlxy, hlyy
def calculate_reduced_hessian_lagrangian(self, hlxx, hlxy, hlyy):
"""
Performs the matrix multiplications necessary to get the
reduced space Hessian-of-Lagrangian term from the full-space
terms.
"""
# Converting to dense is faster for the distillation
# example. Does this make sense?
hlxx = hlxx.toarray()
hlxy = hlxy.toarray()
hlyy = hlyy.toarray()
dydx = self.evaluate_jacobian_external_variables()
term1 = hlxx
prod = hlxy.dot(dydx)
term2 = prod + prod.transpose()
term3 = hlyy.dot(dydx).transpose().dot(dydx)
hess_lag = term1 + term2 + term3
return hess_lag
def evaluate_equality_constraints(self):
return self._nlp.extract_subvector_constraints(self.residual_cons)
def evaluate_jacobian_equality_constraints(self):
nlp = self._nlp
x = self.input_vars
y = self.external_vars
f = self.residual_cons
g = self.external_cons
jfx = nlp.extract_submatrix_jacobian(x, f)
jfy = nlp.extract_submatrix_jacobian(y, f)
jgx = nlp.extract_submatrix_jacobian(x, g)
jgy = nlp.extract_submatrix_jacobian(y, g)
nf = len(f)
nx = len(x)
n_entries = nf * nx
# TODO: Does it make sense to cast dydx to a sparse matrix?
# My intuition is that it does only if jgy is "decomposable"
# in the strongly connected component sense, which is probably
# not usually the case.
dydx = -1 * sps.linalg.splu(jgy.tocsc()).solve(jgx.toarray())
# NOTE: PyNumero block matrices require this to be a sparse matrix
# that contains coordinates for every entry that could possibly
# be nonzero. Here, this is all of the entries.
dfdx = jfx + jfy.dot(dydx)
return _dense_to_full_sparse(dfdx)
def evaluate_jacobian_external_variables(self):
nlp = self._nlp
x = self.input_vars
y = self.external_vars
g = self.external_cons
jgx = nlp.extract_submatrix_jacobian(x, g)
jgy = nlp.extract_submatrix_jacobian(y, g)
jgy_csc = jgy.tocsc()
dydx = -1 * sps.linalg.splu(jgy_csc).solve(jgx.toarray())
return dydx
def evaluate_hessian_external_variables(self):
nlp = self._nlp
x = self.input_vars
y = self.external_vars
g = self.external_cons
jgx = nlp.extract_submatrix_jacobian(x, g)
jgy = nlp.extract_submatrix_jacobian(y, g)
jgy_csc = jgy.tocsc()
jgy_fact = sps.linalg.splu(jgy_csc)
dydx = -1 * jgy_fact.solve(jgx.toarray())
ny = len(y)
nx = len(x)
hgxx = np.array([
get_hessian_of_constraint(con, x, nlp=nlp).toarray() for con in g
])
hgxy = np.array([
get_hessian_of_constraint(con, x, y, nlp=nlp).toarray()
for con in g
])
hgyy = np.array([
get_hessian_of_constraint(con, y, nlp=nlp).toarray() for con in g
])
# This term is sparse, but we do not exploit it.
term1 = hgxx
# This is what we want.
# prod[i,j,k] = sum(hgxy[i,:,j] * dydx[:,k])
prod = hgxy.dot(dydx)
# Swap the second and third axes of the tensor
term2 = prod + prod.transpose((0, 2, 1))
# The term2 tensor could have some sparsity worth exploiting.
# matrix.dot(tensor) is not what we want, so we reverse the order of the
# product. Exploit symmetry of hgyy to only perform one transpose.
term3 = hgyy.dot(dydx).transpose((0, 2, 1)).dot(dydx)
rhs = term1 + term2 + term3
rhs.shape = (ny, nx * nx)
sol = jgy_fact.solve(rhs)
sol.shape = (ny, nx, nx)
d2ydx2 = -sol
return d2ydx2
def evaluate_hessians_of_residuals(self):
"""
This method computes the Hessian matrix of each equality
constraint individually, rather than the sum of Hessians
times multipliers.
"""
nlp = self._nlp
x = self.input_vars
y = self.external_vars
f = self.residual_cons
g = self.external_cons
jfx = nlp.extract_submatrix_jacobian(x, f)
jfy = nlp.extract_submatrix_jacobian(y, f)
dydx = self.evaluate_jacobian_external_variables()
ny = len(y)
nf = len(f)
nx = len(x)
hfxx = np.array([
get_hessian_of_constraint(con, x, nlp=nlp).toarray() for con in f
])
hfxy = np.array([
get_hessian_of_constraint(con, x, y, nlp=nlp).toarray()
for con in f
])
hfyy = np.array([
get_hessian_of_constraint(con, y, nlp=nlp).toarray() for con in f
])
d2ydx2 = self.evaluate_hessian_external_variables()
term1 = hfxx
prod = hfxy.dot(dydx)
term2 = prod + prod.transpose((0, 2, 1))
term3 = hfyy.dot(dydx).transpose((0, 2, 1)).dot(dydx)
d2ydx2.shape = (ny, nx * nx)
term4 = jfy.dot(d2ydx2)
term4.shape = (nf, nx, nx)
d2fdx2 = term1 + term2 + term3 + term4
return d2fdx2
def evaluate_hessian_equality_constraints(self):
"""
This method actually evaluates the sum of Hessians times
multipliers, i.e. the term in the Hessian of the Lagrangian
due to these equality constraints.
"""
# External multipliers must be calculated after both primals and duals
# are set, and are only necessary for this Hessian calculation.
# We know this Hessian calculation wants to use the most recently
# set primals and duals, so we can safely calculate external
# multipliers here.
eq_con_multipliers = self.residual_con_multipliers
self.set_external_constraint_multipliers(eq_con_multipliers)
# These are full-space Hessian-of-Lagrangian terms
hlxx, hlxy, hlyy = self.get_full_space_lagrangian_hessians()
# These terms can be used to calculate the corresponding
# Hessian-of-Lagrangian term in the full space.
hess_lag = self.calculate_reduced_hessian_lagrangian(hlxx, hlxy, hlyy)
sparse = _dense_to_full_sparse(hess_lag)
return sps.tril(sparse)
def set_equality_constraint_scaling_factors(self, scaling_factors):
"""
Set scaling factors for the equality constraints that are exposed
to a solver. These are the "residual equations" in this class.
"""
self.residual_scaling_factors = np.array(scaling_factors)
def get_equality_constraint_scaling_factors(self):
"""
Get scaling factors for the equality constraints that are exposed
to a solver. These are the "residual equations" in this class.
"""
return self.residual_scaling_factors
