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import numpy as np
"""
# 2023 - ported from minpack2.dcsrch, dcstep (Fortran) to Python
c MINPACK-1 Project. June 1983.
c Argonne National Laboratory.
c Jorge J. More' and David J. Thuente.
c
c MINPACK-2 Project. November 1993.
c Argonne National Laboratory and University of Minnesota.
c Brett M. Averick, Richard G. Carter, and Jorge J. More'.
"""
# NOTE this file was linted by black on first commit, and can be kept that way.
class DCSRCH:
"""
Parameters
----------
phi : callable phi(alpha)
Function at point `alpha`
derphi : callable phi'(alpha)
Objective function derivative. Returns a scalar.
ftol : float
A nonnegative tolerance for the sufficient decrease condition.
gtol : float
A nonnegative tolerance for the curvature condition.
xtol : float
A nonnegative relative tolerance for an acceptable step. The
subroutine exits with a warning if the relative difference between
sty and stx is less than xtol.
stpmin : float
A nonnegative lower bound for the step.
stpmax :
A nonnegative upper bound for the step.
Notes
-----
This subroutine finds a step that satisfies a sufficient
decrease condition and a curvature condition.
Each call of the subroutine updates an interval with
endpoints stx and sty. The interval is initially chosen
so that it contains a minimizer of the modified function
psi(stp) = f(stp) - f(0) - ftol*stp*f'(0).
If psi(stp) <= 0 and f'(stp) >= 0 for some step, then the
interval is chosen so that it contains a minimizer of f.
The algorithm is designed to find a step that satisfies
the sufficient decrease condition
f(stp) <= f(0) + ftol*stp*f'(0),
and the curvature condition
abs(f'(stp)) <= gtol*abs(f'(0)).
If ftol is less than gtol and if, for example, the function
is bounded below, then there is always a step which satisfies
both conditions.
If no step can be found that satisfies both conditions, then
the algorithm stops with a warning. In this case stp only
satisfies the sufficient decrease condition.
A typical invocation of dcsrch has the following outline:
Evaluate the function at stp = 0.0d0; store in f.
Evaluate the gradient at stp = 0.0d0; store in g.
Choose a starting step stp.
task = 'START'
10 continue
call dcsrch(stp,f,g,ftol,gtol,xtol,task,stpmin,stpmax,
isave,dsave)
if (task .eq. 'FG') then
Evaluate the function and the gradient at stp
go to 10
end if
NOTE: The user must not alter work arrays between calls.
The subroutine statement is
subroutine dcsrch(f,g,stp,ftol,gtol,xtol,stpmin,stpmax,
task,isave,dsave)
where
stp is a double precision variable.
On entry stp is the current estimate of a satisfactory
step. On initial entry, a positive initial estimate
must be provided.
On exit stp is the current estimate of a satisfactory step
if task = 'FG'. If task = 'CONV' then stp satisfies
the sufficient decrease and curvature condition.
f is a double precision variable.
On initial entry f is the value of the function at 0.
On subsequent entries f is the value of the
function at stp.
On exit f is the value of the function at stp.
g is a double precision variable.
On initial entry g is the derivative of the function at 0.
On subsequent entries g is the derivative of the
function at stp.
On exit g is the derivative of the function at stp.
ftol is a double precision variable.
On entry ftol specifies a nonnegative tolerance for the
sufficient decrease condition.
On exit ftol is unchanged.
gtol is a double precision variable.
On entry gtol specifies a nonnegative tolerance for the
curvature condition.
On exit gtol is unchanged.
xtol is a double precision variable.
On entry xtol specifies a nonnegative relative tolerance
for an acceptable step. The subroutine exits with a
warning if the relative difference between sty and stx
is less than xtol.
On exit xtol is unchanged.
task is a character variable of length at least 60.
On initial entry task must be set to 'START'.
On exit task indicates the required action:
If task(1:2) = 'FG' then evaluate the function and
derivative at stp and call dcsrch again.
If task(1:4) = 'CONV' then the search is successful.
If task(1:4) = 'WARN' then the subroutine is not able
to satisfy the convergence conditions. The exit value of
stp contains the best point found during the search.
If task(1:5) = 'ERROR' then there is an error in the
input arguments.
On exit with convergence, a warning or an error, the
variable task contains additional information.
stpmin is a double precision variable.
On entry stpmin is a nonnegative lower bound for the step.
On exit stpmin is unchanged.
stpmax is a double precision variable.
On entry stpmax is a nonnegative upper bound for the step.
On exit stpmax is unchanged.
isave is an integer work array of dimension 2.
dsave is a double precision work array of dimension 13.
Subprograms called
MINPACK-2 ... dcstep
MINPACK-1 Project. June 1983.
Argonne National Laboratory.
Jorge J. More' and David J. Thuente.
MINPACK-2 Project. November 1993.
Argonne National Laboratory and University of Minnesota.
Brett M. Averick, Richard G. Carter, and Jorge J. More'.
"""
def __init__(self, phi, derphi, ftol, gtol, xtol, stpmin, stpmax):
self.stage = None
self.ginit = None
self.gtest = None
self.gx = None
self.gy = None
self.finit = None
self.fx = None
self.fy = None
self.stx = None
self.sty = None
self.stmin = None
self.stmax = None
self.width = None
self.width1 = None
# leave all assessment of tolerances/limits to the first call of
# this object
self.ftol = ftol
self.gtol = gtol
self.xtol = xtol
self.stpmin = stpmin
self.stpmax = stpmax
self.phi = phi
self.derphi = derphi
def __call__(self, alpha1, phi0=None, derphi0=None, maxiter=100):
"""
Parameters
----------
alpha1 : float
alpha1 is the current estimate of a satisfactory
step. A positive initial estimate must be provided.
phi0 : float
the value of `phi` at 0 (if known).
derphi0 : float
the derivative of `derphi` at 0 (if known).
maxiter : int
Returns
-------
alpha : float
Step size, or None if no suitable step was found.
phi : float
Value of `phi` at the new point `alpha`.
phi0 : float
Value of `phi` at `alpha=0`.
task : bytes
On exit task indicates status information.
If task[:4] == b'CONV' then the search is successful.
If task[:4] == b'WARN' then the subroutine is not able
to satisfy the convergence conditions. The exit value of
stp contains the best point found during the search.
If task[:5] == b'ERROR' then there is an error in the
input arguments.
"""
if phi0 is None:
phi0 = self.phi(0.0)
if derphi0 is None:
derphi0 = self.derphi(0.0)
phi1 = phi0
derphi1 = derphi0
task = b"START"
for i in range(maxiter):
stp, phi1, derphi1, task = self._iterate(
alpha1, phi1, derphi1, task
)
if not np.isfinite(stp):
task = b"WARN"
stp = None
break
if task[:2] == b"FG":
alpha1 = stp
phi1 = self.phi(stp)
derphi1 = self.derphi(stp)
else:
break
else:
# maxiter reached, the line search did not converge
stp = None
task = b"WARNING: dcsrch did not converge within max iterations"
if task[:5] == b"ERROR" or task[:4] == b"WARN":
stp = None # failed
return stp, phi1, phi0, task
def _iterate(self, stp, f, g, task):
"""
Parameters
----------
stp : float
The current estimate of a satisfactory step. On initial entry, a
positive initial estimate must be provided.
f : float
On first call f is the value of the function at 0. On subsequent
entries f should be the value of the function at stp.
g : float
On initial entry g is the derivative of the function at 0. On
subsequent entries g is the derivative of the function at stp.
task : bytes
On initial entry task must be set to 'START'.
On exit with convergence, a warning or an error, the
variable task contains additional information.
Returns
-------
stp, f, g, task: tuple
stp : float
the current estimate of a satisfactory step if task = 'FG'. If
task = 'CONV' then stp satisfies the sufficient decrease and
curvature condition.
f : float
the value of the function at stp.
g : float
the derivative of the function at stp.
task : bytes
On exit task indicates the required action:
If task(1:2) == b'FG' then evaluate the function and
derivative at stp and call dcsrch again.
If task(1:4) == b'CONV' then the search is successful.
If task(1:4) == b'WARN' then the subroutine is not able
to satisfy the convergence conditions. The exit value of
stp contains the best point found during the search.
If task(1:5) == b'ERROR' then there is an error in the
input arguments.
"""
p5 = 0.5
p66 = 0.66
xtrapl = 1.1
xtrapu = 4.0
if task[:5] == b"START":
if stp < self.stpmin:
task = b"ERROR: STP .LT. STPMIN"
if stp > self.stpmax:
task = b"ERROR: STP .GT. STPMAX"
if g >= 0:
task = b"ERROR: INITIAL G .GE. ZERO"
if self.ftol < 0:
task = b"ERROR: FTOL .LT. ZERO"
if self.gtol < 0:
task = b"ERROR: GTOL .LT. ZERO"
if self.xtol < 0:
task = b"ERROR: XTOL .LT. ZERO"
if self.stpmin < 0:
task = b"ERROR: STPMIN .LT. ZERO"
if self.stpmax < self.stpmin:
task = b"ERROR: STPMAX .LT. STPMIN"
if task[:5] == b"ERROR":
return stp, f, g, task
# Initialize local variables.
self.brackt = False
self.stage = 1
self.finit = f
self.ginit = g
self.gtest = self.ftol * self.ginit
self.width = self.stpmax - self.stpmin
self.width1 = self.width / p5
# The variables stx, fx, gx contain the values of the step,
# function, and derivative at the best step.
# The variables sty, fy, gy contain the value of the step,
# function, and derivative at sty.
# The variables stp, f, g contain the values of the step,
# function, and derivative at stp.
self.stx = 0.0
self.fx = self.finit
self.gx = self.ginit
self.sty = 0.0
self.fy = self.finit
self.gy = self.ginit
self.stmin = 0
self.stmax = stp + xtrapu * stp
task = b"FG"
return stp, f, g, task
# in the original Fortran this was a location to restore variables
# we don't need to do that because they're attributes.
# If psi(stp) <= 0 and f'(stp) >= 0 for some step, then the
# algorithm enters the second stage.
ftest = self.finit + stp * self.gtest
if self.stage == 1 and f <= ftest and g >= 0:
self.stage = 2
# test for warnings
if self.brackt and (stp <= self.stmin or stp >= self.stmax):
task = b"WARNING: ROUNDING ERRORS PREVENT PROGRESS"
if self.brackt and self.stmax - self.stmin <= self.xtol * self.stmax:
task = b"WARNING: XTOL TEST SATISFIED"
if stp == self.stpmax and f <= ftest and g <= self.gtest:
task = b"WARNING: STP = STPMAX"
if stp == self.stpmin and (f > ftest or g >= self.gtest):
task = b"WARNING: STP = STPMIN"
# test for convergence
if f <= ftest and abs(g) <= self.gtol * -self.ginit:
task = b"CONVERGENCE"
# test for termination
if task[:4] == b"WARN" or task[:4] == b"CONV":
return stp, f, g, task
# A modified function is used to predict the step during the
# first stage if a lower function value has been obtained but
# the decrease is not sufficient.
if self.stage == 1 and f <= self.fx and f > ftest:
# Define the modified function and derivative values.
fm = f - stp * self.gtest
fxm = self.fx - self.stx * self.gtest
fym = self.fy - self.sty * self.gtest
gm = g - self.gtest
gxm = self.gx - self.gtest
gym = self.gy - self.gtest
# Call dcstep to update stx, sty, and to compute the new step.
# dcstep can have several operations which can produce NaN
# e.g. inf/inf. Filter these out.
with np.errstate(invalid="ignore", over="ignore"):
tup = dcstep(
self.stx,
fxm,
gxm,
self.sty,
fym,
gym,
stp,
fm,
gm,
self.brackt,
self.stmin,
self.stmax,
)
self.stx, fxm, gxm, self.sty, fym, gym, stp, self.brackt = tup
# Reset the function and derivative values for f
self.fx = fxm + self.stx * self.gtest
self.fy = fym + self.sty * self.gtest
self.gx = gxm + self.gtest
self.gy = gym + self.gtest
else:
# Call dcstep to update stx, sty, and to compute the new step.
# dcstep can have several operations which can produce NaN
# e.g. inf/inf. Filter these out.
with np.errstate(invalid="ignore", over="ignore"):
tup = dcstep(
self.stx,
self.fx,
self.gx,
self.sty,
self.fy,
self.gy,
stp,
f,
g,
self.brackt,
self.stmin,
self.stmax,
)
(
self.stx,
self.fx,
self.gx,
self.sty,
self.fy,
self.gy,
stp,
self.brackt,
) = tup
# Decide if a bisection step is needed
if self.brackt:
if abs(self.sty - self.stx) >= p66 * self.width1:
stp = self.stx + p5 * (self.sty - self.stx)
self.width1 = self.width
self.width = abs(self.sty - self.stx)
# Set the minimum and maximum steps allowed for stp.
if self.brackt:
self.stmin = min(self.stx, self.sty)
self.stmax = max(self.stx, self.sty)
else:
self.stmin = stp + xtrapl * (stp - self.stx)
self.stmax = stp + xtrapu * (stp - self.stx)
# Force the step to be within the bounds stpmax and stpmin.
stp = np.clip(stp, self.stpmin, self.stpmax)
# If further progress is not possible, let stp be the best
# point obtained during the search.
if (
self.brackt
and (stp <= self.stmin or stp >= self.stmax)
or (
self.brackt
and self.stmax - self.stmin <= self.xtol * self.stmax
)
):
stp = self.stx
# Obtain another function and derivative
task = b"FG"
return stp, f, g, task
def dcstep(stx, fx, dx, sty, fy, dy, stp, fp, dp, brackt, stpmin, stpmax):
"""
Subroutine dcstep
This subroutine computes a safeguarded step for a search
procedure and updates an interval that contains a step that
satisfies a sufficient decrease and a curvature condition.
The parameter stx contains the step with the least function
value. If brackt is set to .true. then a minimizer has
been bracketed in an interval with endpoints stx and sty.
The parameter stp contains the current step.
The subroutine assumes that if brackt is set to .true. then
min(stx,sty) < stp < max(stx,sty),
and that the derivative at stx is negative in the direction
of the step.
The subroutine statement is
subroutine dcstep(stx,fx,dx,sty,fy,dy,stp,fp,dp,brackt,
stpmin,stpmax)
where
stx is a double precision variable.
On entry stx is the best step obtained so far and is an
endpoint of the interval that contains the minimizer.
On exit stx is the updated best step.
fx is a double precision variable.
On entry fx is the function at stx.
On exit fx is the function at stx.
dx is a double precision variable.
On entry dx is the derivative of the function at
stx. The derivative must be negative in the direction of
the step, that is, dx and stp - stx must have opposite
signs.
On exit dx is the derivative of the function at stx.
sty is a double precision variable.
On entry sty is the second endpoint of the interval that
contains the minimizer.
On exit sty is the updated endpoint of the interval that
contains the minimizer.
fy is a double precision variable.
On entry fy is the function at sty.
On exit fy is the function at sty.
dy is a double precision variable.
On entry dy is the derivative of the function at sty.
On exit dy is the derivative of the function at the exit sty.
stp is a double precision variable.
On entry stp is the current step. If brackt is set to .true.
then on input stp must be between stx and sty.
On exit stp is a new trial step.
fp is a double precision variable.
On entry fp is the function at stp
On exit fp is unchanged.
dp is a double precision variable.
On entry dp is the derivative of the function at stp.
On exit dp is unchanged.
brackt is an logical variable.
On entry brackt specifies if a minimizer has been bracketed.
Initially brackt must be set to .false.
On exit brackt specifies if a minimizer has been bracketed.
When a minimizer is bracketed brackt is set to .true.
stpmin is a double precision variable.
On entry stpmin is a lower bound for the step.
On exit stpmin is unchanged.
stpmax is a double precision variable.
On entry stpmax is an upper bound for the step.
On exit stpmax is unchanged.
MINPACK-1 Project. June 1983
Argonne National Laboratory.
Jorge J. More' and David J. Thuente.
MINPACK-2 Project. November 1993.
Argonne National Laboratory and University of Minnesota.
Brett M. Averick and Jorge J. More'.
"""
sgn_dp = np.sign(dp)
sgn_dx = np.sign(dx)
# sgnd = dp * (dx / abs(dx))
sgnd = sgn_dp * sgn_dx
# First case: A higher function value. The minimum is bracketed.
# If the cubic step is closer to stx than the quadratic step, the
# cubic step is taken, otherwise the average of the cubic and
# quadratic steps is taken.
if fp > fx:
theta = 3.0 * (fx - fp) / (stp - stx) + dx + dp
s = max(abs(theta), abs(dx), abs(dp))
gamma = s * np.sqrt((theta / s) ** 2 - (dx / s) * (dp / s))
if stp < stx:
gamma *= -1
p = (gamma - dx) + theta
q = ((gamma - dx) + gamma) + dp
r = p / q
stpc = stx + r * (stp - stx)
stpq = stx + ((dx / ((fx - fp) / (stp - stx) + dx)) / 2.0) * (stp - stx)
if abs(stpc - stx) <= abs(stpq - stx):
stpf = stpc
else:
stpf = stpc + (stpq - stpc) / 2.0
brackt = True
elif sgnd < 0.0:
# Second case: A lower function value and derivatives of opposite
# sign. The minimum is bracketed. If the cubic step is farther from
# stp than the secant step, the cubic step is taken, otherwise the
# secant step is taken.
theta = 3 * (fx - fp) / (stp - stx) + dx + dp
s = max(abs(theta), abs(dx), abs(dp))
gamma = s * np.sqrt((theta / s) ** 2 - (dx / s) * (dp / s))
if stp > stx:
gamma *= -1
p = (gamma - dp) + theta
q = ((gamma - dp) + gamma) + dx
r = p / q
stpc = stp + r * (stx - stp)
stpq = stp + (dp / (dp - dx)) * (stx - stp)
if abs(stpc - stp) > abs(stpq - stp):
stpf = stpc
else:
stpf = stpq
brackt = True
elif abs(dp) < abs(dx):
# Third case: A lower function value, derivatives of the same sign,
# and the magnitude of the derivative decreases.
# The cubic step is computed only if the cubic tends to infinity
# in the direction of the step or if the minimum of the cubic
# is beyond stp. Otherwise the cubic step is defined to be the
# secant step.
theta = 3 * (fx - fp) / (stp - stx) + dx + dp
s = max(abs(theta), abs(dx), abs(dp))
# The case gamma = 0 only arises if the cubic does not tend
# to infinity in the direction of the step.
gamma = s * np.sqrt(max(0, (theta / s) ** 2 - (dx / s) * (dp / s)))
if stp > stx:
gamma = -gamma
p = (gamma - dp) + theta
q = (gamma + (dx - dp)) + gamma
r = p / q
if r < 0 and gamma != 0:
stpc = stp + r * (stx - stp)
elif stp > stx:
stpc = stpmax
else:
stpc = stpmin
stpq = stp + (dp / (dp - dx)) * (stx - stp)
if brackt:
# A minimizer has been bracketed. If the cubic step is
# closer to stp than the secant step, the cubic step is
# taken, otherwise the secant step is taken.
if abs(stpc - stp) < abs(stpq - stp):
stpf = stpc
else:
stpf = stpq
if stp > stx:
stpf = min(stp + 0.66 * (sty - stp), stpf)
else:
stpf = max(stp + 0.66 * (sty - stp), stpf)
else:
# A minimizer has not been bracketed. If the cubic step is
# farther from stp than the secant step, the cubic step is
# taken, otherwise the secant step is taken.
if abs(stpc - stp) > abs(stpq - stp):
stpf = stpc
else:
stpf = stpq
stpf = np.clip(stpf, stpmin, stpmax)
else:
# Fourth case: A lower function value, derivatives of the same sign,
# and the magnitude of the derivative does not decrease. If the
# minimum is not bracketed, the step is either stpmin or stpmax,
# otherwise the cubic step is taken.
if brackt:
theta = 3.0 * (fp - fy) / (sty - stp) + dy + dp
s = max(abs(theta), abs(dy), abs(dp))
gamma = s * np.sqrt((theta / s) ** 2 - (dy / s) * (dp / s))
if stp > sty:
gamma = -gamma
p = (gamma - dp) + theta
q = ((gamma - dp) + gamma) + dy
r = p / q
stpc = stp + r * (sty - stp)
stpf = stpc
elif stp > stx:
stpf = stpmax
else:
stpf = stpmin
# Update the interval which contains a minimizer.
if fp > fx:
sty = stp
fy = fp
dy = dp
else:
if sgnd < 0:
sty = stx
fy = fx
dy = dx
stx = stp
fx = fp
dx = dp
# Compute the new step.
stp = stpf
return stx, fx, dx, sty, fy, dy, stp, brackt
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