Nonlinear constraint optimization in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of nonlinear optimization with inequality constraint

What is the output of the following code that solves a nonlinear optimization problem with an inequality constraint?

SciPy

from scipy.optimize import minimize

# Objective function: minimize (x-1)^2 + (y-1)^2
fun = lambda v: (v[0]-1)**2 + (v[1]-1)**2

# Inequality constraint: x + y >= 3 (rewritten as x + y - 3 >= 0)
cons = {'type': 'ineq', 'fun': lambda v: v[0] + v[1] - 3}

# Initial guess
x0 = [0, 0]

res = minimize(fun, x0, constraints=cons)
print((round(res.x[0], 2), round(res.x[1], 2)))

A (1.5, 1.5)

B (1.0, 2.0)

C (0.0, 3.0)

D (1.0, 1.0)

💡 Hint

Think about the point closest to (1,1) that satisfies x + y >= 3.

The objective tries to get close to (1,1). The constraint requires x + y >= 3. The closest point on the line x + y = 3 to (1,1) is (1.5,1.5).

❓ data_output

intermediate

Number of iterations in nonlinear constrained optimization

After running this nonlinear optimization with constraints, how many iterations did the solver perform?

SciPy

from scipy.optimize import minimize

fun = lambda v: (v[0]-3)**2 + (v[1]+1)**2
cons = [{'type': 'eq', 'fun': lambda v: v[0]**2 + v[1]**2 - 4}]

x0 = [0, 2]
res = minimize(fun, x0, constraints=cons, options={'disp': False})
print(res.nit)

A 15

B 10

C 5

D 20

💡 Hint

Check the 'nit' attribute in the result object.

The solver converges quickly for this simple problem, performing 5 iterations.

🔧 Debug

advanced

Identify the error in nonlinear optimization constraint definition

What error will this code raise when trying to run nonlinear optimization with constraints?

SciPy

from scipy.optimize import minimize

fun = lambda v: v[0]**2 + v[1]**2
cons = {'type': 'ineq', 'fun': lambda v: v[0] + v[1] - 1, 'jac': lambda v: [1, 1]}

x0 = [0, 0]
res = minimize(fun, x0, constraints=cons)
print(res.fun)

A No error, prints 0.0

B SyntaxError

C TypeError: 'list' object is not callable

D ValueError: jac must be callable or None

💡 Hint

Check the type of 'jac' parameter in constraint dictionary.

The 'jac' parameter must be a callable function returning the Jacobian, not a list.

🧠 Conceptual

advanced

Effect of nonlinear equality constraints on solution space

Which statement best describes the effect of adding a nonlinear equality constraint to an optimization problem?

A It converts the problem into an unconstrained optimization.

B It restricts the solution to a curve or surface where the constraint equals zero.

C It expands the solution space by adding more feasible points.

D It always makes the problem easier to solve.

💡 Hint

Think about what an equality constraint means geometrically.

An equality constraint restricts solutions to points where the constraint function equals zero, often forming a curve or surface.

🚀 Application

expert

Choosing the correct constraint type for a real-world problem

You want to optimize the shape of a container to hold exactly 100 liters of liquid. Which constraint type should you use in scipy.optimize to enforce this volume requirement?

A An inequality constraint specifying volume >= 100

B No constraint needed, just minimize volume

C An inequality constraint specifying volume <= 100

D An equality constraint specifying volume - 100 = 0

💡 Hint

Exact volume means the volume must be equal to 100, not just greater or less.

To enforce an exact volume, use an equality constraint that forces volume - 100 = 0.