/
Minimization.py
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/
Minimization.py
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from __future__ import division
from __future__ import print_function
__author__ = "Maximilian Bisani"
__version__ = "$LastChangedRevision: 1691 $"
__date__ = "$LastChangedDate: 2011-08-03 15:38:08 +0200 (Wed, 03 Aug 2011) $"
__copyright__ = "Copyright (c) 2004-2005 RWTH Aachen University"
__license__ = """
This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License Version 2 (June
1991) 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 General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program; if not, you will find it at
http://www.gnu.org/licenses/gpl.html, or write to the Free Software
Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110,
USA.
Should a provision of no. 9 and 10 of the GNU General Public License
be invalid or become invalid, a valid provision is deemed to have been
agreed upon which comes closest to what the parties intended
commercially. In any case guarantee/warranty shall be limited to gross
negligent actions or intended actions or fraudulent concealment.
"""
from math import *
from numpy import *
gold = (1 + sqrt(5)) / 2
cGold = (3 - sqrt(5)) / 2
def bracketMinimum(f, xa, xb):
"""
Given a unary function f and initial point xa and xb, search in
downhill direction and returns new points xa, xb, xc which bracket
a minimum of f.
adapted from: W. H. Press et. al., "Numerical Recipes", section 10.1
"""
fa = f(xa)
fb = f(xb)
if fb > fa:
xa, xb = xb, xa
fa, fb = fb, fa
xc = xb + gold * (xb - xa)
fc = f(xc)
while fb >= fc:
xuLimit = xb + 100.0 * (xc - xb)
r = (xb - xa) * (fb - fc)
q = (xb - xc) * (fb - fa)
xu = xb - (xb - xc) * q - (xb - xa) * r
if q != r:
xu /= 2 * (q - r)
else:
xu = xuLimit
if (xb - xu) * (xu - xc) > 0.0:
# xu is between xb and xc
fu = f(xu)
if fu < fc:
xa, xb = xb, xu
fa, fb = fb, fu
break
elif fu > fb:
xc = xu
fc = fu
break
xu = xc + gold * (xc - xb)
fu = f(xu)
elif (xc - xu) * (xu - xuLimit) > 0.0:
# xu is between xc and xuLimit
fu = f(xu)
if fu < fc:
xb, xc = xc, xu
fb, fc = fc, fu
xu = xc + gold * (xc - xb)
fu = f(xu)
elif (xu - xuLimit) * (xuLimit - xc) >= 0.0:
xu = xuLimit
fu = f(xu)
else:
xu = xc + gold * (xc - xb)
fu = f(xu)
xa, xb, xc = xb, xc, xu
fa, fb, fc = fb, fc, fu
assert (xa < xb and xb < xc) or (xa > xb and xb > xc)
assert fb <= fa and fb <= fc
return xa, xb, xc, fa, fb, fc
maxIterations = 100
zEpsilon = 1.0e-18
def linearMinimization(
f, x=None, lower=None, upper=None, tolerance=1.0e-10, maxIterations=maxIterations
):
"""
Given a function f and staring point x, this function determines
the minimum of x using Brent's method of parabolic interpolation.
Alternatively lower and upper bounds can be given instead of x.
adapted from: W. H. Press et. al., "Numerical Recipes", section 10.2
"""
if x is not None:
xa, xb, xc, fa, fb, fc = bracketMinimum(f, x, x + 1.0)
if xa < xc:
a, b = xa, xc
else:
a, b = xc, xa
x, fx = xb, fb
elif lower is not None and upper is not None:
a, b = lower, upper
x = a + cGold * (b - a)
fx = f(x)
else:
raise ValueError("Either x or lower and upper must be given.")
d = 0.0
e = 0.0
v, fv = x, fx
w, fw = x, fx
for iteration in range(maxIterations):
xm = (a + b) / 2
tol = tolerance * fabs(x) + zEpsilon
if fabs(x - xm) <= (2.0 * tol - (b - a) / 2):
break
if fabs(e) > tol:
r = (x - w) * (fx - fv)
q = (x - v) * (fx - fw)
p = (x - v) * q - (x - w) * r
q = 2.0 * (q - r)
if q > 0.0:
p = -p
q = fabs(q)
etemp, e = e, d
if fabs(p) >= fabs(0.5 * q * etemp) or p <= q * (a - x) or p >= q * (b - x):
if x >= xm:
e = a - x
else:
e = b - x
d = cGold * e
else:
d = p / q
u = x + d
if u - a < 2.0 * tol or b - u < 2.0 * tol:
if xm >= x:
d = tol
else:
d = -tol
else:
if x >= xm:
e = a - x
else:
e = b - x
d = cGold * e
if fabs(d) > tol:
u = x + d
elif d > 0.0:
u = x + tol
else:
u = x - tol
fu = f(u)
if fu <= fx:
if u >= x:
a = x
else:
b = x
v, w, x = w, x, u
fv, fw, fx = fw, fx, fu
else:
if u < x:
a = u
else:
b = u
if fu < fw or w == x:
v, w = w, u
fv, fw = fw, fu
elif fu <= fv or v == x or v == w:
v = u
fv = fu
else:
raise "failed to converge"
return x, fx
def hasConverged(fCurrent, fOld, tolerance):
return 2 * (fOld - fCurrent) <= tolerance * (fabs(fOld) + fabs(fCurrent) + zEpsilon)
def directionSetMinimization(
f, initialPoint, directions=None, tolerance=1.0e-10, maxIterations=maxIterations
):
"""
Powell's method of multi-dimension minimization.
inspired from: W. H. Press et. al., "Numerical Recipes", section 10.5
"""
if directions is None:
directions = identity(len(initialPoint), type=Float64)
current = initialPoint
fCurrent = f(current)
for iteration in range(maxIterations):
old = current
fOld = fCurrent
largestDecrease = 0.0
directionOfLargestDecrease = None
for dir, dirVector in enumerate(directions):
xMin, fMin = linearMinimization(
lambda x: f(current + x * dirVector), 0, tolerance=tolerance
)
decrease = fCurrent - fMin
if decrease > largestDecrease:
largestDecrease = decrease
directionOfLargestDecrease = dir
current = current + xMin * dirVector
fCurrent = fMin
if fabs(xMin) > zEpsilon:
dirVector *= xMin
if hasConverged(fCurrent, fOld, tolerance):
break
averageDirection = current - old
extrapolated = current + averageDirection
fExtrapolated = f(extrapolated)
if fExtrapolated < fCurrent:
if (
2
* (fOld - 2 * fCurrent + fExtrapolated)
* (fOld - fCurrent - largestDecrease) ** 2
< (fOld - fExtrapolated) ** 2 * largestDecrease
):
directions[directionOfLargestDecrease] = directions[0]
directions[0] = averageDirection
else:
raise "failed to converge"
return current, fCurrent
def hasSignificantDecrease(series):
"""
Determines the slope of a series of values and its standard error.
Returns True if the hypothesis [slope >= 0] can be rejected with
99% confidence.
"""
N = len(series)
x = arange((1 - N) / 2, N / 2)
y = array(series)
assert len(x) == N
assert sum(x) == 0
xx = (N - 1) * N * (N + 1) / 12
assert xx == sum(x * x)
mean = sum(y) / N
slope = sum(x * y) / xx
delta = y - mean - slope * x
if not fabs(sum(delta)) < fabs(mean) * 1e-14:
print("Minimization.py:223:", sum(delta), mean)
sigma = sqrt(sum(delta ** 2) / (N * (N - 1)))
sigmaSlope = sigma / sqrt(xx)
print("b=%f sigma=%f sigma_b=%f" % (slope, sigma, sigmaSlope))
return slope < -2.326348 * sigmaSlope