How to use Scipy Optimize to solve for values when you do not know the closed form 🌱

How does Scipy Optimize Newton Work?

Suppose we want to figure out some values $x_0$ such that the value of some function $f\left(x_0\right)$ approaches $z$.

We can do this with scipy.optimize.newton.

• First, you need to create a loss function which could essentially be the difference between $f\left(x_0\right)$ and $z$ (the function and your desired output)

scipy.optimize.newton(func, x0, fprime=None, args=(), tol=1.48e-08, maxiter=50, fprime2=None, x1=None, rtol=0.0, full_output=False, disp=True)

• The most important inputs you need to run an optimizer is the loss function func and the initial guess for the optimal $x_0$ (i.e. $x_0^{\ast }$)
• Inputting the first and second derivatives allow you have a quicker optimization (by using a different method in the backend of scipy.optimize) but the optimization can still be run (albeit worse performing) without these inputs

Here is an example where we are trying to find some value $x_0$ that when multiplied to the function $e^{2x}log(2x+1)$ will have the mean of the data equal to $7$.

from scipy import optimize

# Dataset
x_series = np.arange(-100,100)
# Value of z
desired_output = 7

def loss_function(x):
	return np.mean(xe^{2x_series}log(2x_series+1)) - desired_output

optimize.newton(loss_function, 0.1)