TopolearnFitting models, calibrating parameters and solving design problems all reduce to minimising a function.
scipy.optimize finds inputs that make a function as small as possible. Maximise by minimising the negative.
For one variable, optimize.minimize_scalar(f) uses Brent's method; pass bounds=(a, b) with method="bounded" to search an interval.
For several variables, optimize.minimize(fun, x0) needs:
fun(x): takes a 1-D array of parameters and returns a float,x0: a starting guess of the same shape,- optionally
args=(...): extra fixed arguments passed tofun, - optionally
method: the default for unconstrained problems is"BFGS";"Nelder-Mead"needs no gradients and tolerates noise;"L-BFGS-B"handles bounds and scales to many variables.
The result is an OptimizeResult. Always check res.success and res.message before using res.x; res.fun is the minimum value and res.nfev counts function evaluations.
These are local optimisers: they walk downhill from x0 and stop in the nearest valley. For non-convex functions, try several starting points or a global method. Scale matters too: if one parameter is around 1e-6 and another around 1e3, rescale them to similar magnitudes or the optimiser struggles.
Gradients speed things up a lot; when you can, pass jac= with a function returning the gradient.
Example
import numpy as np
from scipy import optimize
res1 = optimize.minimize_scalar(lambda x: (x - 2) ** 2 + 1)
print(round(res1.x, 4), round(res1.fun, 4)) # 2.0 1.0
def rosen(p, a=1.0): # classic banana-shaped valley
x, y = p
return (a - x) ** 2 + 100 * (y - x ** 2) ** 2
res = optimize.minimize(rosen, x0=np.array([-1.0, 2.0]), method="BFGS")
print(res.success, res.x.round(4)) # True [1. 1.]
print(res.nfev > 0, round(res.fun, 8)) # True 0.0
res = optimize.minimize(rosen, x0=[0, 0], args=(2.0,), method="Nelder-Mead")
print(res.x.round(3)) # [2. 4.] (minimum moves with a=2)Remember
minimize(fun, x0)withfuntaking a 1-D array and returning a float- Check
res.successbefore trustingres.x - Local optimisers find the nearest minimum; try several starts on non-convex problems
Watch out for
- Returning an array from the objective instead of a single float.