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"""Byrd-Omojokun Trust-Region SQP method."""
from __future__ import division, print_function, absolute_import
from scipy.sparse import eye as speye
from .projections import projections
from .qp_subproblem import modified_dogleg, projected_cg, box_intersections
import numpy as np
from numpy.linalg import norm
__all__ = ['equality_constrained_sqp']
def default_scaling(x):
n, = np.shape(x)
return speye(n)
def equality_constrained_sqp(fun_and_constr, grad_and_jac, lagr_hess,
x0, fun0, grad0, constr0,
jac0, stop_criteria,
state,
initial_penalty,
initial_trust_radius,
factorization_method,
trust_lb=None,
trust_ub=None,
scaling=default_scaling):
"""Solve nonlinear equality-constrained problem using trust-region SQP.
Solve optimization problem:
minimize fun(x)
subject to: constr(x) = 0
using Byrd-Omojokun Trust-Region SQP method described in [1]_. Several
implementation details are based on [2]_ and [3]_, p. 549.
References
----------
.. [1] Lalee, Marucha, Jorge Nocedal, and Todd Plantenga. "On the
implementation of an algorithm for large-scale equality
constrained optimization." SIAM Journal on
Optimization 8.3 (1998): 682-706.
.. [2] Byrd, Richard H., Mary E. Hribar, and Jorge Nocedal.
"An interior point algorithm for large-scale nonlinear
programming." SIAM Journal on Optimization 9.4 (1999): 877-900.
.. [3] Nocedal, Jorge, and Stephen J. Wright. "Numerical optimization"
Second Edition (2006).
"""
PENALTY_FACTOR = 0.3 # Rho from formula (3.51), reference [2]_, p.891.
LARGE_REDUCTION_RATIO = 0.9
INTERMEDIARY_REDUCTION_RATIO = 0.3
SUFFICIENT_REDUCTION_RATIO = 1e-8 # Eta from reference [2]_, p.892.
TRUST_ENLARGEMENT_FACTOR_L = 7.0
TRUST_ENLARGEMENT_FACTOR_S = 2.0
MAX_TRUST_REDUCTION = 0.5
MIN_TRUST_REDUCTION = 0.1
SOC_THRESHOLD = 0.1
TR_FACTOR = 0.8 # Zeta from formula (3.21), reference [2]_, p.885.
BOX_FACTOR = 0.5
n, = np.shape(x0) # Number of parameters
# Set default lower and upper bounds.
if trust_lb is None:
trust_lb = np.full(n, -np.inf)
if trust_ub is None:
trust_ub = np.full(n, np.inf)
# Initial values
x = np.copy(x0)
trust_radius = initial_trust_radius
penalty = initial_penalty
# Compute Values
f = fun0
c = grad0
b = constr0
A = jac0
S = scaling(x)
# Get projections
Z, LS, Y = projections(A, factorization_method)
# Compute least-square lagrange multipliers
v = -LS.dot(c)
# Compute Hessian
H = lagr_hess(x, v)
# Update state parameters
optimality = norm(c + A.T.dot(v), np.inf)
constr_violation = norm(b, np.inf) if len(b) > 0 else 0
cg_info = {'niter': 0, 'stop_cond': 0,
'hits_boundary': False}
last_iteration_failed = False
while not stop_criteria(state, x, last_iteration_failed,
optimality, constr_violation,
trust_radius, penalty, cg_info):
# Normal Step - `dn`
# minimize 1/2*||A dn + b||^2
# subject to:
# ||dn|| <= TR_FACTOR * trust_radius
# BOX_FACTOR * lb <= dn <= BOX_FACTOR * ub.
dn = modified_dogleg(A, Y, b,
TR_FACTOR*trust_radius,
BOX_FACTOR*trust_lb,
BOX_FACTOR*trust_ub)
# Tangential Step - `dt`
# Solve the QP problem:
# minimize 1/2 dt.T H dt + dt.T (H dn + c)
# subject to:
# A dt = 0
# ||dt|| <= sqrt(trust_radius**2 - ||dn||**2)
# lb - dn <= dt <= ub - dn
c_t = H.dot(dn) + c
b_t = np.zeros_like(b)
trust_radius_t = np.sqrt(trust_radius**2 - np.linalg.norm(dn)**2)
lb_t = trust_lb - dn
ub_t = trust_ub - dn
dt, cg_info = projected_cg(H, c_t, Z, Y, b_t,
trust_radius_t,
lb_t, ub_t)
# Compute update (normal + tangential steps).
d = dn + dt
# Compute second order model: 1/2 d H d + c.T d + f.
quadratic_model = 1/2*(H.dot(d)).dot(d) + c.T.dot(d)
# Compute linearized constraint: l = A d + b.
linearized_constr = A.dot(d)+b
# Compute new penalty parameter according to formula (3.52),
# reference [2]_, p.891.
vpred = norm(b) - norm(linearized_constr)
# Guarantee `vpred` always positive,
# regardless of roundoff errors.
vpred = max(1e-16, vpred)
previous_penalty = penalty
if quadratic_model > 0:
new_penalty = quadratic_model / ((1-PENALTY_FACTOR)*vpred)
penalty = max(penalty, new_penalty)
# Compute predicted reduction according to formula (3.52),
# reference [2]_, p.891.
predicted_reduction = -quadratic_model + penalty*vpred
# Compute merit function at current point
merit_function = f + penalty*norm(b)
# Evaluate function and constraints at trial point
x_next = x + S.dot(d)
f_next, b_next = fun_and_constr(x_next)
# Compute merit function at trial point
merit_function_next = f_next + penalty*norm(b_next)
# Compute actual reduction according to formula (3.54),
# reference [2]_, p.892.
actual_reduction = merit_function - merit_function_next
# Compute reduction ratio
reduction_ratio = actual_reduction / predicted_reduction
# Second order correction (SOC), reference [2]_, p.892.
if reduction_ratio < SUFFICIENT_REDUCTION_RATIO and \
norm(dn) <= SOC_THRESHOLD * norm(dt):
# Compute second order correction
y = -Y.dot(b_next)
# Make sure increment is inside box constraints
_, t, intersect = box_intersections(d, y, trust_lb, trust_ub)
# Compute tentative point
x_soc = x + S.dot(d + t*y)
f_soc, b_soc = fun_and_constr(x_soc)
# Recompute actual reduction
merit_function_soc = f_soc + penalty*norm(b_soc)
actual_reduction_soc = merit_function - merit_function_soc
# Recompute reduction ratio
reduction_ratio_soc = actual_reduction_soc / predicted_reduction
if intersect and reduction_ratio_soc >= SUFFICIENT_REDUCTION_RATIO:
x_next = x_soc
f_next = f_soc
b_next = b_soc
reduction_ratio = reduction_ratio_soc
# Readjust trust region step, formula (3.55), reference [2]_, p.892.
if reduction_ratio >= LARGE_REDUCTION_RATIO:
trust_radius = max(TRUST_ENLARGEMENT_FACTOR_L * norm(d),
trust_radius)
elif reduction_ratio >= INTERMEDIARY_REDUCTION_RATIO:
trust_radius = max(TRUST_ENLARGEMENT_FACTOR_S * norm(d),
trust_radius)
# Reduce trust region step, according to reference [3]_, p.696.
elif reduction_ratio < SUFFICIENT_REDUCTION_RATIO:
trust_reduction \
= (1-SUFFICIENT_REDUCTION_RATIO)/(1-reduction_ratio)
new_trust_radius = trust_reduction * norm(d)
if new_trust_radius >= MAX_TRUST_REDUCTION * trust_radius:
trust_radius *= MAX_TRUST_REDUCTION
elif new_trust_radius >= MIN_TRUST_REDUCTION * trust_radius:
trust_radius = new_trust_radius
else:
trust_radius *= MIN_TRUST_REDUCTION
# Update iteration
if reduction_ratio >= SUFFICIENT_REDUCTION_RATIO:
x = x_next
f, b = f_next, b_next
c, A = grad_and_jac(x)
S = scaling(x)
# Get projections
Z, LS, Y = projections(A, factorization_method)
# Compute least-square lagrange multipliers
v = -LS.dot(c)
# Compute Hessian
H = lagr_hess(x, v)
# Set Flag
last_iteration_failed = False
# Otimality values
optimality = norm(c + A.T.dot(v), np.inf)
constr_violation = norm(b, np.inf) if len(b) > 0 else 0
else:
penalty = previous_penalty
last_iteration_failed = True
return x, state
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