Lesson 3.7 - Optimization

Target: ~10 min read - 20 min hands-on

Overview

scipy.optimize.minimize finds the input value(s) that minimize a function you define - perfect for "least material," "lowest cost," or "minimum loss" problems. We'll minimize the sheet-metal area of an open-top cylindrical tank for a required volume.

Why This Matters (Engineering Context)

"Smallest / cheapest design that still meets the requirement" is a constrained minimization whatever the field - a tank, a heat sink, a conductor size, an order quantity. Knowing minimize turns it into a few lines.

Code-Along

from scipy.optimize import minimize
import numpy as np

# [Chemical/Mechanical] minimize the sheet area of an OPEN-TOP cylindrical tank
# for a fixed volume V = pi r^2 h, choosing the radius r (h then follows from V).
V_req = 2.0     # m^3


# The objective: minimize() always passes the variables as an ARRAY, so the
# single radius arrives as r[0]. Return the quantity we want to make smallest.
def surface_area(r):
    r = r[0]
    h = V_req / (np.pi * r ** 2)                  # height implied by the fixed volume
    return np.pi * r ** 2 + 2 * np.pi * r * h     # base area + wall area (no lid)


# x0 = starting guess; bounds keep the search in physically sensible radii
result = minimize(surface_area, x0=[0.5], bounds=[(0.1, 3.0)])
r_opt = result.x[0]                               # result.x holds the optimal variables
h_opt = V_req / (np.pi * r_opt ** 2)

print(f"Optimal radius: {r_opt:.4f} m")
print(f"Height at optimum: {h_opt:.4f} m   (h / r = {h_opt / r_opt:.2f})")
print(f"Minimum sheet area: {result.fun:.4f} m^2")   # result.fun = objective at the optimum
print(f"Converged: {result.success}")

Run it: for an open-top cylinder the optimal proportion is h / r = 1 (height equals radius) - a classic analytic result the optimizer should reproduce to a couple of decimals.

Practice Exercises

1. Change V_req to 5.0 m^3 and re-optimize - do the optimal r and h still satisfy h / r = 1?
2. Add a constraint that the height must not exceed 1.5 m (a headroom limit) and re-solve.
3. Minimize the area for a closed tank instead (add a top: 2 * pi * r**2). What is the optimal h / r now?

# Try the practice exercises here

Knowledge Check

1. What does scipy.optimize.minimize require you to define besides the objective function?
2. What does it mean for a constraint to be "active" or "binding" at the optimum?
3. Why are bounds useful in addition to constraints?