class NonlinearModel:
""" A general nonlinear optimization problem object.
Currently, this generates an IPOPT problem instance through pyipopt.
Eventually it may be desirable to separate the solver interface
from the python representation of the problem, to allow different
solvers, or different interfaces to IPOPT to be used.
Variables and parameters effectively share a namespace with
constraint functions (expand on this later.)
"""
def __init__(self):
"""Initialize an empty model. """
self.constraints = KeyedList()
self.objective_function = None
# List bounds on variables and constraint functions.
# Where upper (lower) bound is not specified, or is None, it is
# treated as positive (negative) infinity.
self.upper_bounds = {}
self.lower_bounds = {}
# Parameters are effectively variables with a fixed value;
# they may be referred to in functions but are not optimized
# over.
self.parameters = {}
# The self.active_nlp flag tracks whether a non-closed pyipopt
# problem instance is attached as self.nlp. This is important because
# self.nlp.close() needs to be called to free up the (often
# substantial) memory used by the problem instance before self.nlp
# is set to a new problem instance (the problems can't be properly
# garbage collected), but attempting to close an already closed
# problem leads to an immediate segmentation fault.
self.active_nlp = False
# Track whether the problem is in a compiled state (lists of
# code objects and constant terms for functions/derivatives,
# variable and constraint counters, etc., are present and
# should be copied, if so.)
self._compiled = False
# Choose which IPOPT option for the handling of variables with
# fixed values we should try to require. Note that this may be
# overridden if the structure of the problem requires it.
# 'make_parameter' is IPOPT's default; I add it here as an
# attribute so that it may be set to other values at need.
self.fixed_variable_treatment = 'make_parameter'
# Allow IPOPT options in general to be set persistently
# for problems associated with this model in an attribute.
# Note at this level we do not distinguish between string,
# integer, and floating-point options; the type of each option
# value will be checked before solving, and the appropriate
# method of the pyipopt nlp instance will be called.
self.ipopt_options = {}
def resolve_name(self, entity_name):
"""Interpret an interactively supplied entity name.
This may be a variable or parameter name, or the name of
a constraint function; not, in general, the name of the objective
function, if that is even defined.
This exists to be overridden by subclasses, allowing friendlier
interactive references to often cumbersome internal variable names.
"""
return entity_name
def get_upper_bound(self, v):
""" Get the upper bound on the variable/constraint v."""
return self.upper_bounds.get(self.resolve_name(v), None)
def get_lower_bound(self, v):
"""Get the lower bound on the variable/constraint v."""
return self.lower_bounds.get(self.resolve_name(v), None)
def get_bounds(self, v):
""" Get a tuple containg bounds on variable/constraint v.
Returns (lower_bound, upper_bound).
"""
return (self.get_lower_bound(v), self.get_upper_bound(v))
def set_upper_bound(self, v, bound):
"""Set the upper bound on the variable/constraint v."""
self.upper_bounds[self.resolve_name(v)] = bound
def set_lower_bound(self, v, bound):
"""Set the lower bound on the variable/constraint v."""
self.lower_bounds[self.resolve_name(v)] = bound
def set_equality(self, v, bound):
""" Set equal lower and upper bounds on variable/constraint v. """
self.set_upper_bound(v, bound)
self.set_lower_bound(v, bound)
def set_inequality(self, v, lower_bound, upper_bound):
""" Set lower and upper bounds on the variable or constant v. """
self.set_upper_bound(v, upper_bound)
self.set_lower_bound(v, lower_bound)
def set_bound(self, v, bound):
""" Set bounds on variable/constraint v with flexible syntax.
If the bound is a scalar or None, it is interpreted as an equality;
if a tuple, as (lower, upper) inequality bounds.
"""
if isinstance(bound, tuple):
self.set_inequality(v, bound[0], bound[1])
else:
self.set_equality(v, bound)
def set_bounds(self, bounds):
""" Set bounds on many variables/constraints from a dict. """
for v, bound in bounds.iteritems():
self.set_bound(v, bound)
def set_objective(self, objective_id, expression):
""" Set the objective function of the problem.
The existing objective function will be lost.
"""
self.objective_function = Function(expression, name=objective_id)
def add_constraint(self, function_id, expression, value=None):
""" Add the expression to the problem as a constraint function.
Optionally, set the value to which the function is constrained,
(a scalar, or (lower bound, upper bound) tuple of scalars or Nones.)
"""
self.constraints.set(function_id, Function(expression,
name=function_id))
if value is not None:
if isinstance(value, tuple):
self.set_inequality(function_id, *value)
else:
self.set_equality(function_id, value)
def remove_constraint(self, function_id):
""" Delete the indicated constraint from the problem.
Any bounds on this constraint will be cleared.
"""
self.constraints.remove_by_key(function_id)
self.upper_bounds.pop(function_id, None)
self.lower_bounds.pop(function_id, None)
def write_bounds_to_file(self, filename):
"""Write current variable and function bounds to a text file.
Each line will contain the entity internal name, its lower bound or
'None', and its upper bound or 'None', separated by tabs.
All entries in self.upper_bounds or self.lower_bounds will
be written to the file, unless both bounds are None, or the key
is not currently a variable or function id.
"""
bound_keys = set(self.lower_bounds.keys())
bound_keys.update(set(self.upper_bounds.keys()))
lines = []
for k in bound_keys:
if k in self.variables or k in self.functions.keys():
l = self.lower_bounds(k)
u = self.upper_bounds(k)
if not (l is None and u is None):
lines.append('\t'.join([k, repr(l), repr(u)]) + '\n')
with open(filename, 'w') as f:
f.write('\n'.join(lines) + '\n')
def make_variable_bound_vectors(self):
""" Prepare vectors of upper and lower bounds on variables.
Returns:
x_L, x_U -- numpy arrays of upper and lower bounds on self.variables
"""
x_L = []
x_U = []
for v in self.variables:
# A variable which is unbounded may be either not included
# in the dictionary of bounds or have a bound of None. In
# either case, we need to give it an 'infinite' bound
# using the effective infinities defined above.
lower_bound = self.get_lower_bound(v)
if lower_bound is None:
lower_bound = IPOPT_MINUS_INF
x_L.append(lower_bound)
upper_bound = self.get_upper_bound(v)
if upper_bound is None:
upper_bound = IPOPT_INF
x_U.append(upper_bound)
return np.array(x_L), np.array(x_U)
def make_constraint_bound_vectors(self):
""" Prepare vectors of upper and lower bounds on constraints.
Returns:
g_L, g_U -- numpy arrays of upper and lower bounds on self.constraints
"""
g_L = []
g_U = []
for constraint in self.constraints.keys():
# As with bounds on variables, unbounded constraint functions
# may either have explicit bounds of None or be excluded
# from the dictionaries of bounds entirely.
lower_bound = self.lower_bounds.get(constraint, None)
if lower_bound is None:
lower_bound = IPOPT_MINUS_INF
g_L.append(lower_bound)
upper_bound = self.upper_bounds.get(constraint, None)
if upper_bound is None:
upper_bound = IPOPT_INF
g_U.append(upper_bound)
g_L = np.array(g_L)
g_U = np.array(g_U)
return g_L, g_U
def correct_guess(self, x0):
"""Project a vector of variable values inside the bounds.
This may be useful for example in setting up an initial guess from
which to start the optimizer. Note that IPOPT should automatically
project all variables inside their bounds at initialization, but
this has proven helpful nonetheless, eg in situations where
some bounds have been relaxed by IPOPT (due to fixed_variable_treatment
relax_bounds.)
This function considers only consistency with bounds
on variables, not satisfaction of equality/inequality constraints.
Arguments:
x0 -- list or array of length of self.variables
Returns:
x1 -- modified copy of x0.
"""
x1 = x0.copy()
for i, x in enumerate(x0):
l, u = self.get_bounds(self.variables[i])
if l is not None and x < l:
x1[i] = l
if u is not None and x > u:
x1[i] = u
return x1
#######
# CORE IPOPT INTERFACE CODE
def _update_namespace(self, x):
""" Fill self._namespace with the values of variables, parameters at x. """
self._namespace = dict(zip(self.variables, x))
self._namespace.update(self.parameters)
def _eval(self, code):
""" Evaluate code object (or expression) at the current point.
The code is evaluated in the context of nlcm._common_namespace and
self._namespace.
"""
return eval(code, _common_namespace, self._namespace)
def update_variables(self):
"""Find the decision variables of the problem and order them.
All the functions in the problem (constraints and objective)
are searched for the variables on which they depend, and those
which are in self.parameters are excluded.
The variables are then placed, sorted, into the tuple
self.variables, and the dictionary self.var_index is set up to
map a variable string to its index in the tuple. The variable
counter s.nvar is updated.
"""
variables = set()
for f in self.constraints:
variables.update(f.variables)
# faster, if less reliable, than
# variables.update(em.extract_vars(f.math))
variables.update(self.objective_function.variables)
variables = variables - set(self.parameters)
# Use a tuple to make clear that this ordering should not be
# modified casually.
variables = tuple(sorted(variables))
self.variables = variables
self.nvar = len(variables)
var_index = {}
for i, v in enumerate(variables):
var_index[v] = i
self.var_index = var_index
def compile(self):
"""Compile objective, constraint, and derivative functions.
Lists of compiled code objects and constants representing
variable and constant terms in the objective and constraint
functions and their first and second derivatives, indices of
nonzero terms, etc., are prepared for use by the methods
self.eval_f, self.eval_g, etc. Miscellaneous attributes
(e.g. the counters of variables, constraints, and nonzero Jacobian
and Hessian elements, self.nvar, self.ncon, self._nnzj, self._nnzh)
are set.
"""
self.update_variables() # also sets self.nvar
# Set up the objective and its gradient
self.setup_f()
self.setup_grad_f()
# Set up constraints and derivatives
self.setup_g()
self.ncon = len(self.constraints)
self.setup_jac_g() # also sets self._nnzj
# Set up the Hessian.
self.setup_h() # also sets self._nnzh
self._compiled = True
def setup_f(self):
""" Set up the objective function, ensuring its code is compiled. """
# Calling the code() method compiles the function and caches the result,
# if this has not been done already.
self.objective_function.code()
def eval_f(self, x):
""" Evaluate the objective function at point x. """
self._update_namespace(x)
return self._eval(self.objective_function.code())
def setup_grad_f(self):
"""Find constant and variable terms of the gradient of the objective.
This method populates the (self.nvar,) array
self._grad_f_constant_terms with the values of any constant
terms of the gradient (it is otherwise zero) and the list
self._grad_f_variable_terms with tuples (index,
code_object_for_that_index_in_grad_f).
"""
constant_terms = np.zeros(self.nvar)
variable_terms = []
all_variables = set(self.variables)
derivatives = self.objective_function.all_first_derivatives(
all_variables)
for variable, derivative in derivatives.iteritems():
variable_index = self.var_index[variable]
if em.extract_vars(derivative):
code = self.objective_function.first_derivative_code(variable)
variable_terms.append((variable_index, code))
else:
constant_terms[variable_index] = np.float(derivative)
self._grad_f_constant_terms = constant_terms
self._grad_f_variable_terms = variable_terms
def eval_grad_f(self, x):
""" Evaluate the gradient of the objective function.
Returns an array of shape (self.nvar,) containing the gradient
at x.
"""
self._update_namespace(x)
gradient = self._grad_f_constant_terms.copy()
for i, code in self._grad_f_variable_terms:
gradient[i] += self._eval(code)
return gradient
def setup_g(self):
""" Set up constraint functions by ensuring their code is compiled. """
for constraint_function in self.constraints:
constraint_function.code()
def eval_g(self, x):
self._update_namespace(x)
return np.array(
[self._eval(constraint.code()) for constraint in self.constraints],
dtype=np.float)
def setup_jac_g(self):
"""List nonzero derivatives of the constraints, corresponding code objects.
We determine the (constraint_index, variable_index) pairs
corresponding to (potentially) nonzero constraint derivatives
and list them in self._jac_g_indices, setting self._nnzj to
the length of this list. Then, the array of length nnzj
self._jac_g_constant_terms is initialized with the numerical
values of the terms which are constant (elsewhere zero), and
self._jac_g_variable_terms is set to a list of (index,
code_object) tuples, where the indices are the positions of
the variable terms in the overall list of nozero Jacobian
terms.
"""
indices = []
constant_terms = []
variable_terms = []
all_variables = set(self.variables)
term = 0 # index of the current nonzero term
for i, constraint in enumerate(self.constraints):
derivatives = constraint.all_first_derivatives(all_variables)
for variable, expression in derivatives.iteritems():
j = self.var_index[variable]
indices.append((i, j))
if em.extract_vars(expression):
code = constraint.first_derivative_code(variable)
variable_terms.append((term, code))
constant_terms.append(0.)
else:
constant_terms.append(expression_to_float(expression))
term += 1
self._nnzj = len(indices)
self._jac_g_indices = indices
self._jac_g_constant_terms = np.array(constant_terms, dtype=np.float)
self._jac_g_variable_terms = variable_terms
def eval_jac_g(self, x, flag=False):
""" Evaluate or list nonzero elements of the constraint Jacobian.
Arguments:
x - numpy array of length self.nvar
flag - Boolean indicating whether the indices of the nonzero elements
should be returned.
Returns: if flag is not set, a numpy array of the values of the
(possibly) nonzero elements of the Jacobian at x. If flag is set,
a 2-tuple of numpy arrays, the first giving the constraints
and the second the variables associated with the nonzero Jacobian
elements, is returned.
"""
if flag:
return tuple(map(np.array, zip(*self._jac_g_indices)))
self._update_namespace(x)
gradient = self._jac_g_constant_terms.copy()
for k, code in self._jac_g_variable_terms:
gradient[k] += self._eval(code)
return gradient
def setup_h(self):
"""Identify nonzero second derivatives of objective and constraints.
Pairs of variables with respect to which the second derivatives of
any such function are not (necessarily) zero are identified and a
corresponding list of pairs of indices (i,j), i>j, into the upper
triangular part of the Hessian is stored as self._h_indices.
self._nnzh is set to the length of this list.
The lists self._h_objective_constant_terms and
self._h_objective_variable_terms are populated with tuples
(relevant_index_into_self.h_indices, term) where the term is
either a numpy float or a compiled code object as appropriate.
The lists self._h_constraint_variable_terms and
self._h_constraint_constant_terms are populated with tuples
(relevant_index_into_self.h_indices,
constraint_index,
term)
where the term is a float or code object as appropriate.
"""
all_variables = set(self.variables)
objective_terms = self.objective_function.all_second_derivatives(
all_variables)
constraint_terms = {
i: constraint.all_second_derivatives(all_variables)
for i, constraint in enumerate(self.constraints)
}
keys = set(objective_terms)
for i, derivatives in constraint_terms.iteritems():
keys.update(derivatives)
self._nnzh = len(keys)
key_to_index_pair = {
(v1, v2): tuple(sorted((self.var_index[v1], self.var_index[v2])))
for v1, v2 in keys
}
self._h_indices = sorted(key_to_index_pair.values())
index_pair_to_h_index = dict([
(pair, i) for i, pair in enumerate(self._h_indices)
])
key_to_h_index = {
k: index_pair_to_h_index[key_to_index_pair[k]]
for k in keys
}
objective_constant_terms = []
objective_variable_terms = []
for k, expression in objective_terms.iteritems():
if em.extract_vars(expression):
code = self.objective_function.second_derivative_code(k)
objective_variable_terms.append((key_to_h_index[k], code))
else:
objective_constant_terms.append(
(key_to_h_index[k], expression_to_float(expression)))
self._h_objective_constant_terms = objective_constant_terms
self._h_objective_variable_terms = objective_variable_terms
constraint_constant_terms = []
constraint_variable_terms = []
for i, terms in constraint_terms.iteritems():
for k, expression in terms.iteritems():
if em.extract_vars(expression):
code = self.constraints[i].second_derivative_code(k)
constraint_variable_terms.append(
(key_to_h_index[k], i, code))
else:
constraint_constant_terms.append(
(key_to_h_index[k], i,
expression_to_float(expression)))
self._h_constraint_constant_terms = constraint_constant_terms
self._h_constraint_variable_terms = constraint_variable_terms
def eval_h(self, x, lagrange, obj_factor, flag=None):
"""Evaluate or list nonzero elements of the Hessian.
Specifically, what is evaluated is the Hessian of the Lagrangian
in the particular, slightly unconventional, form described in
eqn. 9 of the section "Interfacing with IPOPT through code" of the
IPOPT online documentation.
x - numpy array of length self.nvar
lagrange - array of length self.ncon giving Lagrange multipliers
for the constraints
obj_factor - float, used to scale the contributions of the
objective function to the Hessian
flag - Boolean indicating whether the indices of the nonzero elements
should be returned.
Returns: if flag is not set, a numpy array of the values of the
(possibly) nonzero elements of the Hessian at x, appropriately
scaled by the Lagrange multipliers and objective function factor.
If flag is set, a 2-tuple of numpy arrays, giving first
and second indices respectively for the possibly nonzero
entries of the upper triangular part of the Hessian matrix, is returned.
"""
# This could be sped up in various ways, most notably if the
# objective factor or Lagrange multipliers are sometimes
# exactly zero, or close enough that we are comfortable
# treating them as zero.
if flag:
return tuple(map(np.array, zip(*self._h_indices)))
self._update_namespace(x)
h = np.zeros(self._nnzh)
# First populate with the terms related to the objective,
# unscaled.
if self._h_objective_constant_terms:
indices, values = zip(*self._h_objective_constant_terms)
# indices is a tuple, but numpy array indexing treats
# tuples differently from lists; we want an index
# array rather than a multi-dimensional index, so
# we cast to list explicitly.
h[list(indices)] = values
for index, code in self._h_objective_variable_terms:
h[index] = self._eval(code)
# Then scale by the objective factor
h *= obj_factor
# Then add the terms associated with the constraints, scaling
# by the Lagrange multipliers
for index, constraint_index, value in self._h_constraint_constant_terms:
h[index] += lagrange[constraint_index] * value
for index, constraint_index, code in self._h_constraint_variable_terms:
h[index] += lagrange[constraint_index] * self._eval(code)
return h
def repeated_solve(self, x0, max_iter, max_attempts, options={}):
""" Solve the nonlinear problem, restarting as needed.
In some cases it is advantageous to stop the progress of the solver
and restart it from the same point, resetting the internal variable
scaling. (There are probably better ways to do this, but this works
in practice.) Here, the solver will be restarted if it exits with
status -1 (iteration limit exceeded) or -2 (restoration failure).
Arguments:
x0 -- starting point
max_iter -- how many iterations to run before restarting (passed
to IPOPT as max_iter)
max_attempts -- how many times to restart before giving up
options -- additional options for the solve() method.
"""
all_options = {'max_iter': max_iter}
all_options.update(options)
attempt = 0
x = x0
while attempt < max_attempts:
try:
x = self.solve(x0=x, options=all_options)
except OptimizationFailure as e:
if self.status not in {-1, -2}:
raise e
else:
x = self.x.copy()
else:
return x
attempt += 1
# We have made the maximum number of restart attempts
# but still not converged, or still ended up with a restoration
# failure.
raise OptimizationFailure(
'No convergence after %d attempts (status %d)' %
(max_attempts, self.status))
def solve(self, x0=None, options={}):
"""Solve the nonlinear problem from a specified starting point.
Arguments:
x0 -- array of length self.nvars with initial guesses for values of
self.variables (in that order), or a scalar to be used as the initial
guess for all variables; if not supplied, all variables will be
set initially to 1.0.
options -- dictionary {'option': value} of options to pass to
IPOPT. Options will be taken from (in increasing order of priority)
IPOPT's default values, the module-wide default_ipopt_options, those
specified in self.fixed_variable_treatment or self.ipopt_options,
this argument, or the ipopt.opt file (if any) in the working directory.
Errors will result if invalid names or values for options are given
here.
Returns:
x -- array of variable values returned by IPOPT (ordered as in
self.variables)
Also sets self.x, self.zl, self.zu, self.constraint_multipliers,
self.obj_value, and self.status to x, the lower and upper
bound multipliers, the Lagrange multipliers associated with
the constraint functions, the final objective function value,
and the optimzation status, respectively, and sets self.soln
to a dictionary mapping each variable key to its value in
self.x.
This method does not recompile the objective, constraint and
derivative functions if self.compile() has already been
called. If anything except bounds on variables, bounds on
constraints, and/or parameter values has been changed since
the last time self.compile() has been called, self.compile
must be called again before solving, or this method will
attempt to solve an out-of-date version of the problem.
Each call to solve does create a new pyipopt problem instance
as self.nlp, closing the old one beforehand if self.active_nlp
is true, and resetting self.active_nlp to true afterwards.
"""
if not self._compiled:
self.compile()
if x0 is None:
x0 = np.ones(self.nvar)
elif np.isscalar(x0):
x0 = x0 * np.ones(self.nvar)
else:
# Assume this is a vector. Check its length, as trying to
# proceed with an inappropriately sized starting point may
# lead to crashes.
if len(x0) != self.nvar:
message = (
'Starting point has wrong dimension (needed %d, given %d)'
% (self.nvar, len(x0)))
raise ValueError(message)
self.close_nlp()
x_L, x_U = self.make_variable_bound_vectors()
g_L, g_U = self.make_constraint_bound_vectors()
# The following avoids a segmentation fault that results
# when bound methods are supplied as evaluation functions
# to pyipopt, which I don't really understand:
eval_f = lambda x, user_data=None: self.eval_f(x)
eval_grad_f = lambda x, user_data=None: self.eval_grad_f(x)
eval_g = lambda x, user_data=None: self.eval_g(x)
eval_jac_g = lambda x, flag, user_data=None: self.eval_jac_g(x, flag)
eval_h= lambda x, lagrange, obj_factor, flag, user_data=None: \
self.eval_h(x,lagrange,obj_factor,flag)
self.nlp = pyipopt.create(self.nvar, x_L, x_U, self.ncon, g_L, g_U,
self._nnzj, self._nnzh, eval_f, eval_grad_f,
eval_g, eval_jac_g, eval_h)
self.active_nlp = True
# Handle generic IPOPT options
all_options = {}
all_options.update(default_ipopt_options)
all_options['fixed_variable_treatment'] = self.fixed_variable_treatment
all_options.update(self.ipopt_options)
all_options.update(options)
for option, value in all_options.iteritems():
if isinstance(value, str):
self.nlp.str_option(option, value)
elif isinstance(value, int):
self.nlp.int_option(option, value)
else:
self.nlp.num_option(option, value)
(self.x, self.zl, self.zu, self.constraint_multipliers, self.obj_value,
self.status) = self.nlp.solve(x0)
self.soln = dict(zip(self.variables, self.x))
if self.status not in (0, 1):
raise OptimizationFailure('IPOPT exited with status %d' %
self.status)
return self.x.copy()
def close_nlp(self):
"""Close the nonlinear programming instance if one appears to be open.
If self.active_nlp is True, call sell self.nlp.close() and reset
self.active_nlp to False; otherwise pass. Possibly a warning ought
to be issued, as if the nlp instance really is open somewhere,
failure to close it will leak memory, but at the moment this
is omitted to avoid spurious warnings when setting the object's
first nlp instance.
"""
if self.active_nlp:
self.nlp.close()
self.active_nlp = False
# END CORE IPOPT INTERFACE CODE
######
def __getstate__(self):
""" Return a dict of attributes for copying, excluding the NLP. """
state = self.__dict__.copy()
if 'nlp' in state:
del state['nlp']
state['active_nlp'] = False
return state
def copy(self):
""" Return a (deep) copy of the NLP instance. """
return copy.deepcopy(self)