# Copyright 2021 IRT Saint Exupéry, https://www.irt-saintexupery.com
#
# This program is free software; you can redistribute it and/or
# modify it under the terms of the GNU Lesser General Public
# License version 3 as published by the Free Software Foundation.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
# Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with this program; if not, write to the Free Software Foundation,
# Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
# Contributors:
# INITIAL AUTHORS - API and implementation and/or documentation
# :author: Pierre-Jean Barjhoux
# :author: Francois Gallard, integration and cleanup
# OTHER AUTHORS - MACROSCOPIC CHANGES
"""Implementation of the Lagrange multipliers."""
from __future__ import annotations
import logging
from numpy import arange
from numpy import array
from numpy import atleast_2d
from numpy import concatenate
from numpy import ndarray
from numpy import where
from numpy import zeros
from numpy.linalg import matrix_rank
from numpy.linalg import norm
from scipy.optimize import linprog
from scipy.optimize import nnls
from gemseo.algos.design_space import DesignSpace
from gemseo.algos.opt_problem import OptimizationProblem
from gemseo.third_party.prettytable import PrettyTable
LOGGER = logging.getLogger(__name__)
[docs]class LagrangeMultipliers:
r"""Class that implements the computation of Lagrange Multipliers.
Denote :math:`x^\ast` an optimal solution of the optimization problem
below.
.. math::
\begin{aligned}
& \text{Minimize} & & f(x) \\
& \text{relative to} & & x \\
& \text{subject to} & & \left\{\begin{aligned}
& g(x)\le0, \\
& h(x)=0, \\
& \ell\le x\le u.
\end{aligned}\right.
\end{aligned}
If the constraints are qualified at :math:`x^\ast` then the Lagrange
multipliers of :math:`x^\ast` are the vectors :math:`\lambda_g`,
:math:`\lambda_h`, :math:`\lambda_\ell` and :math:`\lambda_u` satisfying
.. math::
\left\{\begin{aligned}
&\frac{\partial f}{\partial x}(x^\ast)
+\lambda_g^\top\frac{\partial g}{\partial x}(x^\ast)
+\lambda_h^\top\frac{\partial h}{\partial x}(x^\ast)
+\sum_j\lambda_{\ell,j}+\sum_j\lambda_{u,j}
=0,\\
&\lambda_{g,i}\ge0\text{ if }g_i(x^\ast)=0,
\text{ otherwise }\lambda_{g,i}=0,\\
&\lambda_{\ell,j}\le0\text{ if }x^\ast_j=\ell_j,
\text{ otherwise }\lambda_{\ell,j}=0,\\
&\lambda_{u,j}\ge0\text{ if }x^\ast_j=u_j,
\text{ otherwise }\lambda_{u,j}=0.
\end{aligned}\right.
"""
LOWER_BOUNDS = "lower_bounds"
UPPER_BOUNDS = "upper_bounds"
INEQUALITY = "inequality"
EQUALITY = "equality"
CSTR_LABELS = [LOWER_BOUNDS, UPPER_BOUNDS, INEQUALITY, EQUALITY]
def __init__(self, opt_problem: OptimizationProblem) -> None:
"""
Args:
opt_problem: The optimization problem
on which Lagrange multipliers shall be computed.
"""
self._check_inputs(opt_problem)
self.opt_problem = opt_problem
self.active_lb_names = []
self.active_ub_names = []
self.active_ineq_names = []
self.active_eq_names = []
self.lagrange_multipliers = None
self.__normalized = opt_problem.preprocess_options.get(
"is_function_input_normalized", False
)
@staticmethod
def _check_inputs(opt_problem: OptimizationProblem) -> None:
"""Verify that an problem is an instance of :class:`.OptimizationProblem`.
Args:
opt_problem: The optimization problem
on which Lagrange multipliers shall be computed.
Raises:
ValueError: When the problem is not an :class:`.OptimizationProblem`
or when the problem was not solved.
"""
if not isinstance(opt_problem, OptimizationProblem):
raise ValueError(
"LagrangeMultipliers must be initialized with an OptimizationProblem."
)
if opt_problem.solution is None:
raise ValueError("The optimization problem was not solved.")
[docs] def compute(
self, x_vect: ndarray, ineq_tolerance: float = 1e-6, rcond: float = -1
) -> dict[str, tuple[list[str], ndarray]]:
"""Compute the Lagrange multipliers, as a post-processing of the optimal point.
This solves:
(d ActiveConstraints)' d Objective
(-------------------) . Lambda = - -----------
(d X ) d X
Args:
x_vect: The optimal point on which the multipliers shall be computed.
ineq_tolerance: The tolerance on inequality constraints.
rcond: The cut-off ratio for small singular values of the Jacobian
(see scipy.linalg.lsq).
Returns:
The Lagrange multipliers.
"""
LOGGER.info("Computation of Lagrange multipliers")
# Check feasibility
self._check_feasibility(x_vect)
# get jacobian of all active constraints, and an
# ordered list of their name
jac_act, _ = self._get_jac_act(x_vect, ineq_tolerance)
if jac_act is None:
# There is no active constraint
multipliers = []
self._store_multipliers(multipliers)
return self.lagrange_multipliers
lhs = jac_act.T
act_constr_nb = lhs.shape[1]
rank = matrix_rank(lhs)
LOGGER.info("Found %s active constraints", str(act_constr_nb))
LOGGER.info("Rank of jacobian = %s", str(rank))
if act_constr_nb > rank:
LOGGER.warning("Number of active constraints > rank !")
# get jacobian of objective
obj_jac = self._get_obj_jac(x_vect)
rhs = -obj_jac.T
# Compute the Lagrange multipliers as a feasible solution of a
# linear optimization problem
act_eq_constr_nb = len(self.active_eq_names)
bounds = [(0, None)] * (act_constr_nb - act_eq_constr_nb) + [
(None, None)
] * act_eq_constr_nb
optim_result = linprog(zeros(act_constr_nb), A_eq=lhs, b_eq=rhs, bounds=bounds)
if optim_result.status == 2:
LOGGER.warning("Lagrange multipliers appear not to exist")
if optim_result.success:
mul = optim_result.x
else:
# If the linear optimization failed then obtain the Lagrange
# multipliers as a solution of a least-square problem
mul, residuals = nnls(lhs, rhs)
LOGGER.info("Residuals norm = %s", str(norm(residuals)))
# stores multipliers in a dictionary
self._store_multipliers(mul)
return self.lagrange_multipliers
def _check_feasibility(self, x_vect: ndarray) -> None:
"""Check that the given point is in the design space and satisfies all
constraints.
Args:
x_vect: The point at which the Lagrange multipliers are to be computed.
"""
problem = self.opt_problem
# Check that the point is within bounds
problem.design_space.check_membership(x_vect)
# Check that the point satisfies other constraints
values, _ = problem.evaluate_functions(
x_vect, eval_obj=False, normalize=False, no_db_no_norm=True
)
feasible = problem.is_point_feasible(values)
if not feasible:
LOGGER.warning("Infeasible point, Lagrange multipliers may not exist.")
def _get_act_bound_jac(self, act_bounds: dict[str, ndarray]):
"""Return the Jacobian of the active bounds.
The constraints sign is not taken into account (matrix is made of 0 and 1).
Args:
act_bounds: The active bounds.
Returns:
The Jacobian of the active bounds
and the name of each component of each function.
"""
dspace = self.opt_problem.design_space
x_dim = dspace.dimension
dim_act = sum(len(where(bnd)[0]) for bnd in act_bounds.values())
if dim_act == 0:
return None, []
act_array = concatenate([act_bounds[var] for var in dspace.variables_names])
bnd_jac = zeros((dim_act, x_dim))
if self.__normalized:
norm_factor = dspace.get_upper_bounds() - dspace.get_lower_bounds()
act_jac = norm_factor[act_array]
else:
act_jac = 1.0
bnd_jac[arange(dim_act), act_array] = act_jac
indexed_varnames = array(dspace.get_indexed_variables_names())
act_b_names = indexed_varnames[act_array].tolist()
return bnd_jac, act_b_names
def __get_act_ineq_jac(
self, x_vect: ndarray, ineq_tolerance: float = 1e-6
) -> tuple[ndarray, list[str]]:
"""Return the Jacobian of the active inequality constraints.
Args:
x_vect: The point at which the Jacobian is computed.
ineq_tolerance: The tolerance for the inequality constraints.
Returns:
The Jacobian of the active inequality constraints
and the name of each component of each function.
"""
# retrieves the active functions and the indices :
# a function is active if at least
# one of its component (in case of multidimensional constraints) is
# active
act_func = self.opt_problem.get_active_ineq_constraints(x_vect, ineq_tolerance)
dspace = self.opt_problem.design_space
if self.__normalized:
x_vect = dspace.normalize_vect(x_vect)
jac = []
names = []
for func, act_set in act_func.items():
if True in act_set:
ineq_jac = func.jac(x_vect)
if len(ineq_jac.shape) == 1:
# Make sure the Jacobian is a 2-dimensional array
ineq_jac = ineq_jac.reshape((1, x_vect.size))
else:
ineq_jac = ineq_jac[act_set, :]
jac.append(ineq_jac)
if func.dim == 1:
names.append(func.name)
else:
names += [
self._get_component_name(func.name, i)
for i, active in enumerate(act_set)
if active
]
if jac:
jac = concatenate(jac)
else:
jac = None
return jac, names
def _get_act_eq_jac(self, x_vect: ndarray) -> tuple[ndarray, list[str]]:
"""Return The Jacobian of the active equality constraints.
Args:
x_vect: The point at which the Jacobian is computed.
Returns:
The Jacobian of the active equality constraints
and the name of each component of each function.
"""
eq_functions = self.opt_problem.get_eq_constraints()
# loop on equality functions
# NB: as the solution (x_vect) is supposed to be feasible,
# all functions (on all dimensions) are supposed to be active
jac = []
names = []
dspace = self.opt_problem.design_space
if self.__normalized:
x_vect = dspace.normalize_vect(x_vect)
for eq_function in eq_functions:
eq_jac = atleast_2d(eq_function.jac(x_vect))
jac.append(eq_jac)
if eq_function.dim == 1:
names.append(eq_function.name)
else:
names += [
self._get_component_name(eq_function.name, i)
for i in range(eq_jac.shape[0])
]
if jac:
jac = concatenate(jac)
else:
jac = None
return jac, names
def _get_obj_jac(self, x_vect: ndarray) -> ndarray:
"""Return the Jacobian of the objective.
Args:
x_vect: The point at which the Jacobian is computed.
Returns:
The Jacobian of the objective.
"""
if self.__normalized:
x_vect = self.opt_problem.design_space.normalize_vect(x_vect)
return self.opt_problem.objective.jac(x_vect)
def _get_jac_act(
self, x_vect: ndarray, ineq_tolerance: float = 1e-6
) -> tuple[ndarray, list[str]]:
"""Return the Jacobian of the active constraints.
Args:
x_vect: The point at which the Jacobian is computed.
ineq_tolerance: The tolerance for the inequality constraints.
Returns:
The Jacobian of the active constraints
and the name of each component of each function.
"""
# Bounds jacobian
dspace = self.opt_problem.design_space
act_lb, act_ub = dspace.get_active_bounds(x_vect, tol=ineq_tolerance)
lb_jac_act, self.active_lb_names = self._get_act_bound_jac(act_lb)
if lb_jac_act is not None:
lb_jac_act *= -1
ub_jac_act, self.active_ub_names = self._get_act_bound_jac(act_ub)
# inequality names
tol = ineq_tolerance
ineq_jac, self.active_ineq_names = self.__get_act_ineq_jac(x_vect, tol)
# equality names
eq_jac, eq_names_act = self._get_act_eq_jac(x_vect)
self.active_eq_names = eq_names_act
names = (
self.active_lb_names
+ self.active_ub_names
+ self.active_ineq_names
+ eq_names_act
)
jacobians = [
jacobian
for jacobian in [lb_jac_act, ub_jac_act, ineq_jac, eq_jac]
if jacobian is not None
]
if jacobians:
jac_act_arr = concatenate(jacobians, axis=0)
else:
# There no active constraint
jac_act_arr = None
return jac_act_arr, names
def _store_multipliers(self, multipliers: ndarray) -> None:
"""Store the Lagrange multipliers in the attribute :attr:`lagrange_multipliers`.
Args:
multipliers: The Lagrange multipliers.
"""
lag = {}
i_min = 0
n_act = len(self.active_lb_names)
if n_act > 0:
l_b_mult = multipliers[i_min : i_min + n_act]
lag[self.LOWER_BOUNDS] = (self.active_lb_names, l_b_mult)
i_min += n_act
wrong_inds = where(l_b_mult < 0.0)[0]
if wrong_inds.size > 0:
names_neg = array(self.active_lb_names)[wrong_inds]
LOGGER.warning(
"Negative Lagrange multipliers for "
"lower bounds on variables"
"%s !",
str(names_neg),
)
n_act = len(self.active_ub_names)
if n_act > 0:
u_b_mult = multipliers[i_min : i_min + n_act]
lag[self.UPPER_BOUNDS] = (self.active_ub_names, u_b_mult)
i_min += n_act
wrong_inds = where(u_b_mult < 0.0)[0]
if wrong_inds.size > 0:
names_neg = array(self.active_ub_names)[wrong_inds]
LOGGER.warning(
"Negative Lagrange multipliers for "
"upper bounds on variables"
"%s !",
str(names_neg),
)
n_act = len(self.active_ineq_names)
if n_act > 0:
ineq_mult = multipliers[i_min : i_min + n_act]
lag[self.INEQUALITY] = (self.active_ineq_names, ineq_mult)
i_min += n_act
wrong_inds = where(ineq_mult < 0.0)[0]
if wrong_inds.size > 0:
names_neg = array(self.active_ineq_names)[wrong_inds]
LOGGER.warning(
"Negative Lagrange multipliers for "
"inequality constraints"
"%s !",
str(names_neg),
)
n_act = len(self.active_eq_names)
if n_act > 0:
lag[self.EQUALITY] = (
self.active_eq_names,
multipliers[i_min : i_min + n_act],
)
i_min += n_act
self.lagrange_multipliers = lag
def _initialize_multipliers(self) -> dict[str, dict[str, ndarray]]:
"""Initialize the Lagrange multipliers with zeros.
Returns:
The Lagrange multipliers.
"""
problem = self.opt_problem
multipliers = dict()
# Bound-constraints
indexed_varnames = problem.design_space.get_indexed_variables_names()
multipliers[self.LOWER_BOUNDS] = dict.fromkeys(indexed_varnames, 0.0)
multipliers[self.UPPER_BOUNDS] = dict.fromkeys(indexed_varnames, 0.0)
# Inequality-constraints
multipliers[self.INEQUALITY] = {
func.name if func.dim == 1 else self._get_component_name(func.name, i): 0.0
for func in problem.get_ineq_constraints()
for i in range(func.dim)
}
# Equality-constraints
multipliers[self.EQUALITY] = {
func.name if func.dim == 1 else self._get_component_name(func.name, i): 0.0
for func in problem.get_eq_constraints()
for i in range(func.dim)
}
return multipliers
[docs] def get_multipliers_arrays(self) -> dict[str, dict[str, ndarray]]:
"""Return the Lagrange multipliers (zero and nonzero) as arrays.
Returns:
The Lagrange multipliers.
"""
problem = self.opt_problem
design_space = problem.design_space
# Convert to dictionaries
multipliers = dict()
for label in self.CSTR_LABELS:
names, mults = self.lagrange_multipliers.get(label, ([], array([])))
multipliers[label] = dict(zip(names, mults))
# Add the Lagrange multipliers equal to zero
multipliers_init = self._initialize_multipliers()
for label in self.CSTR_LABELS:
multipliers_init[label].update(multipliers[label])
# Cast the multipliers as arrays
mult_arrays = dict()
# Bound-constraints multipliers
mult_arrays[self.LOWER_BOUNDS] = dict()
mult_arrays[self.UPPER_BOUNDS] = dict()
for name in design_space.variables_names:
indexed_varnames = design_space.get_indexed_variables_names()
var_low_mult = array(
[
multipliers_init[self.LOWER_BOUNDS][comp_name]
for comp_name in indexed_varnames
]
)
mult_arrays[self.LOWER_BOUNDS][name] = var_low_mult
var_upp_mult = array(
[
multipliers_init[self.UPPER_BOUNDS][comp_name]
for comp_name in indexed_varnames
]
)
mult_arrays[self.UPPER_BOUNDS][name] = var_upp_mult
# Inequality-constraints multipliers
ineq_mult = multipliers_init[self.INEQUALITY]
mult_arrays[self.INEQUALITY] = dict()
for func in problem.get_ineq_constraints():
func_mult = array(
[
ineq_mult[
func.name
if func.dim == 1
else self._get_component_name(func.name, index)
]
for index in range(func.dim)
]
)
mult_arrays[self.INEQUALITY][func.name] = func_mult
# Equality-constraints multipliers
eq_mult = multipliers_init[self.EQUALITY]
mult_arrays[self.EQUALITY] = dict()
for func in problem.get_eq_constraints():
func_mult = array(
[
eq_mult[
func.name
if func.dim == 1
else self._get_component_name(func.name, index)
]
for index in range(func.dim)
]
)
mult_arrays[self.EQUALITY][func.name] = func_mult
return mult_arrays
@staticmethod
def _get_component_name(name: str, index: int) -> str:
"""Return the name of a variable component.
Args:
name: The name of the variable.
index: The index of the component.
Returns:
The name of the variable component.
"""
return f"{name}{DesignSpace.SEP}{index}"
def _get_pretty_table(self) -> PrettyTable:
"""Display the Lagrange Multipliers."""
table = PrettyTable(
["Constraint type", "Constraint name", "Lagrange Multiplier"]
)
for cstr_type, nam_val in self.lagrange_multipliers.items():
for name, value in zip(nam_val[0], nam_val[1]):
table.add_row([cstr_type, name, value])
return table
def __str__(self, *args, **kwargs) -> str:
return f"Lagrange multipliers:\n{self._get_pretty_table().get_string()}"
