scipy.optimize.leastsq with bound constraints
Solution 1
scipy.optimize.least_squares in scipy 0.17 (January 2016) handles bounds; use that, not this hack.
Bound constraints can easily be made quadratic,
and minimized by leastsq along with the rest.
Say you want to minimize a sum of 10 squares Σ f_i(p)^2,
so your func(p) is a 10-vector [f0(p) ... f9(p)],
and also want 0 <= p_i <= 1 for 3 parameters.
Consider the "tub function" max( - p, 0, p - 1 ),
which is 0 inside 0 .. 1 and positive outside, like a \_____/ tub.
If we give leastsq
the 13-long vector
[ f0(p), f1(p), ... f9(p), w*tub(p0), w*tub(p1), w*tub(p2) ]
with w = say 100, it will minimize the sum of squares of the lot:
the tubs will constrain 0 <= p <= 1.
General lo <= p <= hi is similar.
The following code is just a wrapper that runs leastsq
with e.g. such a 13-long vector to minimize.
# leastsq_bounds.py
# see also test_leastsq_bounds.py on gist.github.com/denis-bz
from __future__ import division
import numpy as np
from scipy.optimize import leastsq
__version__ = "2015-01-10 jan denis" # orig 2012
#...............................................................................
def leastsq_bounds( func, x0, bounds, boundsweight=10, **kwargs ):
""" leastsq with bound conatraints lo <= p <= hi
run leastsq with additional constraints to minimize the sum of squares of
[func(p) ...]
+ boundsweight * [max( lo_i - p_i, 0, p_i - hi_i ) ...]
Parameters
----------
func() : a list of function of parameters `p`, [err0 err1 ...]
bounds : an n x 2 list or array `[[lo_0,hi_0], [lo_1, hi_1] ...]`.
Use e.g. [0, inf]; do not use NaNs.
A bound e.g. [2,2] pins that x_j == 2.
boundsweight : weights the bounds constraints
kwargs : keyword args passed on to leastsq
Returns
-------
exactly as for leastsq,
http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.leastsq.html
Notes
-----
The bounds may not be met if boundsweight is too small;
check that with e.g. check_bounds( p, bounds ) below.
To access `x` in `func(p)`, `def func( p, x=xouter )`
or make it global, or `self.x` in a class.
There are quite a few methods for box constraints;
you'll maybe sing a longer song ...
Comments are welcome, test cases most welcome.
"""
# Example: test_leastsq_bounds.py
if bounds is not None and boundsweight > 0:
check_bounds( x0, bounds )
if "args" in kwargs: # 8jan 2015
args = kwargs["args"]
del kwargs["args"]
else:
args = ()
#...............................................................................
funcbox = lambda p: \
np.hstack(( func( p, *args ),
_inbox( p, bounds, boundsweight )))
else:
funcbox = func
return leastsq( funcbox, x0, **kwargs )
def _inbox( X, box, weight=1 ):
""" -> [tub( Xj, loj, hij ) ... ]
all 0 <=> X in box, lo <= X <= hi
"""
assert len(X) == len(box), \
"len X %d != len box %d" % (len(X), len(box))
return weight * np.array([
np.fmax( lo - x, 0 ) + np.fmax( 0, x - hi )
for x, (lo,hi) in zip( X, box )])
# def tub( x, lo, hi ):
# """ \___/ down to lo, 0 lo .. hi, up from hi """
# return np.fmax( lo - x, 0 ) + np.fmax( 0, x - hi )
#...............................................................................
def check_bounds( X, box ):
""" print Xj not in box, loj <= Xj <= hij
return nr not in
"""
nX, nbox = len(X), len(box)
assert nX == nbox, \
"len X %d != len box %d" % (nX, nbox)
nnotin = 0
for j, x, (lo,hi) in zip( range(nX), X, box ):
if not (lo <= x <= hi):
print "check_bounds: x[%d] %g is not in box %g .. %g" % (j, x, lo, hi)
nnotin += 1
return nnotin
Solution 2
scipy has several constrained optimization routines in scipy.optimize. The constrained least squares variant is scipy.optimize.fmin_slsqp
Solution 3
The capability of solving nonlinear least-squares problem with bounds, in an optimal way as mpfit does, has long been missing from Scipy.
This much-requested functionality was finally introduced in Scipy 0.17, with the new function scipy.optimize.least_squares.
This new function can use a proper trust region algorithm to deal with bound constraints, and makes optimal use of the sum-of-squares nature of the nonlinear function to optimize.
Notes:
The solution proposed by @denis has the major problem of introducing a discontinuous "tub function". This renders the scipy.optimize.leastsq
optimization, designed for smooth functions, very inefficient, and possibly unstable, when the boundary is crossed.
The use of scipy.optimize.minimize
with method='SLSQP'
(as @f_ficarola suggested) or scipy.optimize.fmin_slsqp
(as @matt suggested), have the major problem of not making use of the sum-of-square nature of the function to be minimized. These functions are both designed to minimize scalar functions (true also for fmin_slsqp, notwithstanding the misleading name). These approaches are less efficient and less accurate than a proper one can be.
Solution 4
Have a look at: http://lmfit.github.io/lmfit-py/, it should solve your problem.
user1293231
Updated on November 01, 2020Comments
-
user1293231 over 3 years
I am looking for an optimisation routine within scipy/numpy which could solve a non-linear least-squares type problem (e.g., fitting a parametric function to a large dataset) but including bounds and constraints (e.g. minima and maxima for the parameters to be optimised). At the moment I am using the python version of mpfit (translated from idl...): this is clearly not optimal although it works very well.
An efficient routine in python/scipy/etc could be great to have ! Any input is very welcome here :-)
thanks!