Minimizing multivariate functions (minimize) in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of scipy minimize with Nelder-Mead

What is the output of the following code snippet that minimizes a simple quadratic function using Nelder-Mead method?

SciPy

import numpy as np
from scipy.optimize import minimize

def f(x):
    return (x[0] - 1)**2 + (x[1] + 2)**2

result = minimize(f, x0=[0, 0], method='Nelder-Mead')
print(result.x.round(2))

A [0.00, 0.00]

B [-1.00, 2.00]

C [1.00, 2.00]

D [1.00, -2.00]

💡 Hint

Think about where the function (x-1)^2 + (y+2)^2 is smallest.

The function is minimized when x=1 and y=-2, so the minimize function finds this point.

❓ data_output

intermediate

Number of iterations in BFGS minimization

How many iterations does the BFGS method take to minimize the Rosenbrock function starting at [-1.2, 1]?

SciPy

import numpy as np
from scipy.optimize import minimize

def rosenbrock(x):
    return 100*(x[1] - x[0]**2)**2 + (1 - x[0])**2

result = minimize(rosenbrock, x0=[-1.2, 1], method='BFGS')
print(result.nit)

A 26

B 100

C 50

D 10

💡 Hint

The Rosenbrock function is tricky; BFGS usually takes a few dozen iterations.

The BFGS method converges in 26 iterations for this starting point.

🔧 Debug

advanced

Identify the error in minimize call

What error does this code raise when trying to minimize a function with constraints?

SciPy

from scipy.optimize import minimize

def f(x):
    return x[0]**2 + x[1]**2

cons = ({'type': 'eq', 'fun': lambda x: x[0] + x[1] - 1})

result = minimize(f, x0=[0, 0], constraints=cons)
print(result.success)

A TypeError: 'dict' object is not iterable

B ValueError: Constraints must be a sequence

C No error, prints True

D SyntaxError

💡 Hint

Check the type of the constraints argument.

Constraints must be a list or tuple of dicts, not a single dict.

🧠 Conceptual

advanced

Effect of initial guess on minimize

Which statement best describes the effect of the initial guess on the result of scipy.optimize.minimize for non-convex functions?

A The initial guess can lead to different local minima being found.

B The initial guess does not affect the result; global minimum is always found.

C The initial guess only affects the speed, not the final solution.

D The initial guess must be the exact minimum to succeed.

💡 Hint

Think about hills and valleys in a bumpy landscape.

For non-convex functions, minimize may find different local minima depending on the start point.

🚀 Application

expert

Minimize a function with bounds and constraints

Which option correctly minimizes the function f(x,y) = (x-2)^2 + (y-3)^2 with the constraint x + y = 5 and bounds 0 <= x <= 3, 0 <= y <= 4?

SciPy

from scipy.optimize import minimize

def f(x):
    return (x[0] - 2)**2 + (x[1] - 3)**2

cons = {'type': 'eq', 'fun': lambda x: x[0] + x[1] - 5}
bounds = [(0, 3), (0, 4)]

result = minimize(f, x0=[0, 0], constraints=[cons], bounds=bounds, method='SLSQP')
print(result.x.round(2))

A [2.00, 3.00]

B [1.00, 4.00]

C [3.00, 2.00]

D [0.00, 5.00]

💡 Hint

The constraint forces x + y = 5, and bounds limit x and y.

The minimum inside bounds and constraint is at x=3, y=2, which satisfies x+y=5 and bounds.