Root Finding & Optimization (scipy.optimize)

Optimization and root finding answer the questions: “Where does my system reach equilibrium?” and “What parameter values give the lowest cost or smallest error?“

1. 1D Root Finding: brentq vs newton

When you have a scalar function $f(x) = 0$:

from scipy.optimize import brentq
root = brentq(f, a, b)

from scipy.optimize import newton
root = newton(f, x0=1.0, fprime=df_dx)

from scipy.optimize import root
sol = root(system_func, x0=[1.0, 1.0])

Example: Solving Transcendental Roots

Solve $x \cdot e^x = 1$ (the Omega constant, analytical solution via Lambert W function):

2. Multivariable Minimization (minimize)

scipy.optimize.minimize handles both unconstrained and constrained optimization problems:

Example: Constrained Tank Optimization

Design a cylindrical tank of fixed volume $V = 1.0\text{ dm}^3$ with minimal surface area $A = 2\pi r^2 + 2\pi r h$. Analytically, the optimum occurs at $h = 2r \implies r = (V / 2\pi)^{1/3} \approx 0.5419\text{ dm}$:

3. Linear Programming (linprog)

scipy.optimize.linprog solves problems of the form $\min \mathbf{c}^T \mathbf{x}$ subject to $A_{ub} \mathbf{x} \le \mathbf{b}_{ub}$.

Suppose a machine shop produces two items:

• Profit: Item 1 = $40, Item 2 = $30.
• Constraints: Machine hours $\le 12$ ($x_1 + x_2 \le 12$), Raw material $\le 16\text{ kg}$ ($2x_1 + x_2 \le 16$). We maximize profit (minimize $-40x_1 - 30x_2$):

4. Nonlinear Curve Fitting (curve_fit)

scipy.optimize.curve_fit estimates parameters $\boldsymbol{\theta}$ for non-linear regression $y = f(x; \boldsymbol{\theta})$ via Levenberg-Marquardt or Trust Region Reflective algorithms, returning both parameter estimates and the parameter covariance matrix pcov:

5. Try It Yourself

Problem: Minimize a Quadratic Function

Find the minimum of $f(x) = (x - 3.5)^2 + 1.25$ using scipy.optimize.minimize.

from scipy.optimize import minimize
 
def f(x):
    return (x[0] - 3.5)**2 + 1.25
 
res = minimize(f, x0=[0.0], method='BFGS')
print(f"Optimal x: {res.x[0]:.4f}")
print(f"Minimum f(x): {res.fun:.4f}")