Bounds and constraints in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of constrained optimization with bounds

What is the value of result.x after running this code that minimizes f(x) = (x-3)^2 with bounds 0 <= x <= 2?

SciPy

from scipy.optimize import minimize

f = lambda x: (x - 3)**2
bounds = [(0, 2)]
result = minimize(f, x0=[0], bounds=bounds)
print(result.x)

A [1.5]

B [3.]

C [0.]

D [2.]

💡 Hint

Think about the minimum of the function without bounds and how bounds limit the solution.

The function (x-3)^2 has its minimum at x=3. But the bounds restrict x to be between 0 and 2. So the optimizer picks the closest bound, which is 2.

❓ data_output

intermediate

Number of iterations with linear constraint

How many iterations does the optimizer perform when minimizing f(x) = (x[0]-1)^2 + (x[1]-2)^2 with constraint x[0] + x[1] = 3 and no bounds?

SciPy

from scipy.optimize import minimize

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

A 2

B 1

C 3

D 0

💡 Hint

The problem is simple and quadratic with a linear equality constraint.

The optimizer converges quickly because the problem is convex and simple, requiring only one iteration.

🔧 Debug

advanced

Identify error in constraint definition

What error does this code raise when trying to minimize f(x) = x[0]^2 + x[1]^2 with constraint x[0] - 1 > 0 defined incorrectly?

SciPy

from scipy.optimize import minimize

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

A ValueError

B SyntaxError

C TypeError

D No error, runs successfully

💡 Hint

Check what the constraint function returns.

The constraint function returns a boolean value instead of a numeric value, causing a TypeError during optimization.

🚀 Application

advanced

Choosing bounds for parameter estimation

You want to estimate parameters a and b for a model y = a * x + b using least squares. You know a must be positive and b between -5 and 5. Which bounds should you use in scipy.optimize.minimize?

A bounds=[(0, None), (-5, 5)]

B bounds=[(None, 0), (-5, 5)]

C bounds=[(0, None), (None, 5)]

D bounds=[(0, 5), (-5, 5)]

💡 Hint

None means no limit in that direction.

To enforce a > 0, use (0, None). For b between -5 and 5, use (-5, 5).

🧠 Conceptual

expert

Effect of bounds and constraints on solution uniqueness

Which statement is true about the effect of bounds and constraints on the uniqueness of the solution in optimization problems?

A Adding bounds or constraints can make a unique solution non-unique by restricting the feasible region.

B Bounds and constraints always guarantee a unique solution by limiting the search space.

C Bounds and constraints always make the problem unsolvable.

D Bounds and constraints never affect the uniqueness of the solution.

💡 Hint

Think about how restricting the feasible region can create multiple optimal points.

Adding bounds or constraints can reduce the feasible region and sometimes create multiple solutions with the same optimal value, losing uniqueness.