Constrained optimization in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of constrained minimization with inequality

What is the minimum value found by this constrained optimization code using scipy.optimize.minimize?

SciPy

from scipy.optimize import minimize

# Objective function
f = lambda x: (x[0]-1)**2 + (x[1]-2.5)**2

# Constraint: x0 + x1 >= 2
cons = ({'type': 'ineq', 'fun': lambda x: x[0] + x[1] - 2})

# Initial guess
x0 = [2, 0]

res = minimize(f, x0, constraints=cons)
print(round(res.fun, 4))

A 0.25

B 0.0

C 1.25

D 2.25

💡 Hint

Think about the point closest to (1, 2.5) that satisfies x0 + x1 >= 2.

The minimum is at (1,1), which satisfies the constraint x0 + x1 = 2. The function value is (1-1)^2 + (1-2.5)^2 = 0 + 2.25 = 2.25, but since the initial guess is (2,0), the optimizer finds (1.25, 0.75) with value 0.25.

❓ data_output

intermediate

Number of iterations in constrained optimization

How many iterations does the following constrained optimization take to converge?

SciPy

from scipy.optimize import minimize

f = lambda x: (x[0]-3)**2 + (x[1]+1)**2
cons = ({'type': 'eq', 'fun': lambda x: x[0] - 2*x[1]})
x0 = [0, 0]
res = minimize(f, x0, constraints=cons)
print(res.nit)

A 7

B 3

C 10

D 5

💡 Hint

Check the number of iterations attribute in the result object.

The solver converges in 5 iterations for this simple quadratic with linear equality constraint.

🔧 Debug

advanced

Identify the error in constraint definition

What error will this code raise when running scipy.optimize.minimize with the given constraints?

SciPy

from scipy.optimize import minimize

f = lambda x: x[0]**2 + x[1]**2
cons = ({'type': 'ineq', 'fun': lambda x: x[0] - 1},
        {'type': 'eq', 'fun': lambda x: x[1] + 2})
x0 = [0, 0]
res = minimize(f, x0, constraints=cons)
print(res.fun)

A TypeError: 'tuple' object is not callable

B ValueError: Constraints must be a dict or list of dicts

C RuntimeWarning: Initial guess does not satisfy constraints

D No error, outputs 1.0

💡 Hint

Check if the initial guess satisfies the constraints.

The initial guess x0 = [0,0] does not satisfy the inequality x0[0] - 1 >= 0 (0 - 1 = -1 < 0) and equality x0[1] + 2 = 2 != 0, so the solver warns about initial guess violation.

❓ visualization

advanced

Plot of feasible region and minimum point

Which plot correctly shows the feasible region and the minimum point for the problem: Minimize f(x,y) = (x-2)^2 + (y-3)^2 Subject to constraints: x + y <= 4 and x >= 0, y >= 0?

SciPy

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-1,5,400)
y = np.linspace(-1,5,400)
X, Y = np.meshgrid(x,y)
Z = (X-2)**2 + (Y-3)**2

plt.contour(X, Y, Z, levels=30)
plt.fill_between(x, 0, 4 - x, where=(x<=4), color='lightblue', alpha=0.5)
plt.xlim(-1,5)
plt.ylim(-1,5)
plt.scatter(1.5, 2.5, color='red', label='Minimum')
plt.legend()
plt.show()

A Plot with shaded square region from (0,0) to (4,4) with red dot at (2,3)

B Plot with shaded triangle region bounded by x=0, y=0, and x+y=4 with red dot at (1.5, 2.5)

C Plot with shaded circle region centered at (2,3) with red dot at (2,3)

D Plot with no shading and red dot at (0,0)

💡 Hint

The feasible region is where x and y are non-negative and their sum is at most 4.

The constraints form a triangle bounded by x=0, y=0, and x+y=4. The minimum point inside this region is at (1.5, 2.5), which is the closest point to (2,3) inside the feasible area.

🚀 Application

expert

Interpreting Lagrange multipliers from constrained optimization

After solving the constrained optimization problem: Minimize f(x,y) = x^2 + y^2 Subject to x + 2y = 6 The solver returns Lagrange multiplier value 3. What does this multiplier represent?

A The rate of increase in the minimum value of f when the constraint constant increases by 1

B The minimum value of the function f at the solution point

C The value of x at the solution point

D The number of iterations the solver took

💡 Hint

Lagrange multipliers measure sensitivity of the optimal value to constraint changes.

The Lagrange multiplier shows how much the minimum value of the objective function would increase if the right side of the constraint (6) increased by 1 unit.