scipy.optimize.leastsq with bound constraints

Question

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

Answer 1

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.

Answer 2

scipy has several constrained optimization routines in scipy.optimize. The constrained least squares variant is scipy.optimize.fmin_slsqp

Answer 3

Have a look at: http://lmfit.github.io/lmfit-py/, it should solve your problem.

Answer 4

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