Convex constrained-optimization

Introduction to Optimization in Python

Jasmin Ludolf

Content Developer

Convex constrained-optimization

Convex function.

Constraint: limitation on variables
Indifference curve: represents a combination of variables
Find the point on the indifference curve that minimizes or maximizes the objective function

The indifference curve

A young woman with dark hair works on a laptop

Freelance software engineer
Values work ($w$) and leisure ($l$)

Utility function:

$U(w, l)=w^{0.4}l^{0.6}$

Plotting the indifference curve

import numpy as np
import matplotlib.pyplot as plt


w = np.linspace(1, 30, 100)  
l = np.linspace(1, 30, 100)

W, L = np.meshgrid(w, l)

F = W**0.4 * L**0.6


plt.figure(figsize=(8, 6))

contours = plt.contour(W, L, F, levels=[5, 10, 15, 20])

plt.clabel(contours)

plt.title('Indifference Curves for the function: w**0.4 * l**0.6')
plt.xlabel('w')
plt.ylabel('l')
plt.grid(True)
plt.show()

The indifference curve

Visualizing the indifference curve

Alexia is indifferent about the specific combination of $w$ and $l$

Time constraint

24 hours per day
Linear equality constraint

$$ \max_{w,l} w^{0.4}l^{0.6}$$ $$s.t.\ \ \ w+l = 24$$

Add constraint to the graph:
- l = 24 - w
- plt.plot(w, l, color='red')

Indifference curve and linear constraint. Indifference curve at optimal level is tangent to constraint. At the tangency point the gradient is perpendicular to the constraint and the tangent.

Solving with SciPy

def utility_function(vars):
    w, l = vars
    return -(w**0.4 * l**0.6)


def constraint(vars):
    return 24 - np.sum(vars)


initial_guess = [12, 12]
constraint_definition = {'type': 'eq', 'fun': constraint}
result = minimize(utility_function, initial_guess, constraints=constraint_definition)
print(result.x)

[ 9.60001122 14.39998878]

Let's practice!

Introduction to Optimization in Python