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.

Practice

1. What is the main purpose of scipy.optimize.minimize in data science?

easy

A. To find the minimum value of a multivariate function

B. To sort data in ascending order

C. To calculate the mean of a dataset

D. To generate random numbers

Solution

1. Step 1: Understand the function's goal

   scipy.optimize.minimize is designed to find the lowest point of a function with multiple variables.

2. Step 2: Compare options with the function's purpose

   Sorting, calculating mean, or generating random numbers are unrelated to minimizing functions.

3. Final Answer:

   To find the minimum value of a multivariate function -> Option A

4. Quick Check:

   Minimize means find lowest point [OK]

Hint: Minimize means find lowest point of a function [OK]

Common Mistakes:

• Confusing minimize with sorting or statistics
• Thinking it generates random data
• Assuming it calculates averages

2. Which of the following is the correct way to call scipy.optimize.minimize for a function f with initial guess x0?

easy

A. minimize(f, x0, method='max')

B. minimize(f, x0)

C. minimize(f(x0))

D. minimize(x0, f)

Solution

1. Step 1: Recall the function signature

   The correct call is minimize(fun, x0) where fun is the function and x0 is the starting guess.

2. Step 2: Check each option

   minimize(f, x0) matches the correct order. minimize(x0, f) reverses arguments. minimize(f(x0)) calls the function instead of passing it. minimize(f, x0, method='max') uses an invalid method 'max'.

3. Final Answer:

   minimize(f, x0) -> Option B

4. Quick Check:

   Function first, guess second [OK]

Hint: Pass function first, then initial guess [OK]

Common Mistakes:

• Swapping function and initial guess
• Calling function instead of passing it
• Using invalid method names

3. What will be the output of this code snippet?

from scipy.optimize import minimize

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

result = minimize(f, [0, 0])
print(result.x.round(2))

medium

A. [1.00, 2.00]

B. [0.00, 0.00]

C. [-1.00, 2.00]

D. [1.00, -2.00]

Solution

1. Step 1: Understand the function to minimize

   The function is (x0-1)^2 + (x1+2)^2, which is smallest when x0=1 and x1=-2.

2. Step 2: Check the minimize result

   Starting from [0,0], minimize finds the point closest to the minimum, which is [1, -2].

3. Final Answer:

   [1.00, -2.00] -> Option D

4. Quick Check:

   Minimum at x=1, y=-2 [OK]

Hint: Minimum at points making squared terms zero [OK]

Common Mistakes:

• Confusing signs inside the function
• Expecting initial guess as output
• Misreading the function formula

4. Identify the error in this code that tries to minimize a function:

from scipy.optimize import minimize

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

result = minimize(f, 0)
print(result.x)

medium

A. Initial guess should be a list or array, not a single number

B. Function f should return a list, not a number

C. minimize requires method='BFGS' explicitly

D. print statement should be inside a function

Solution

1. Step 1: Check the initial guess type

   The function expects a vector input like [x0, x1], but the initial guess is a single number 0, which is invalid.

2. Step 2: Understand the correct input format

   Initial guess must be a list or array with the same dimension as the function input, e.g., [0, 0].

3. Final Answer:

   Initial guess should be a list or array, not a single number -> Option A

4. Quick Check:

   Input vector must match function variables [OK]

Hint: Initial guess must match function input shape [OK]

Common Mistakes:

• Passing scalar instead of vector as initial guess
• Thinking function output must be list
• Assuming method must be specified always

5. You want to minimize the function f(x, y) = (x-3)^2 + (y+1)^2 but only allow y to be between -2 and 0. Which code correctly applies this constraint using scipy.optimize.minimize?

hard

A. minimize(lambda v: (v[0]-3)**2 + (v[1]+1)**2, [0, 0], constraints={'type':'ineq', 'fun': lambda v: v[1] + 2})

B. minimize(lambda v: (v[0]-3)**2 + (v[1]+1)**2, [0, 0], bounds=[(-2, 0), (None, None)])

C. minimize(lambda v: (v[0]-3)**2 + (v[1]+1)**2, [0, 0], bounds=[(None, None), (-2, 0)])

D. minimize(lambda v: (v[0]-3)**2 + (v[1]+1)**2, [0, 0], constraints={'type':'eq', 'fun': lambda v: v[1] + 1})

Solution

1. Step 1: Understand bounds parameter

   Bounds are given as a list of (min, max) tuples for each variable. Here, x has no bounds, y is between -2 and 0.

2. Step 2: Match bounds to variables

   minimize(lambda v: (v[0]-3)**2 + (v[1]+1)**2, [0, 0], bounds=[(None, None), (-2, 0)]) correctly sets bounds as [(None, None), (-2, 0)] meaning x unbounded, y bounded. minimize(lambda v: (v[0]-3)**2 + (v[1]+1)**2, [0, 0], bounds=[(-2, 0), (None, None)]) reverses bounds. Options A and D use constraints incorrectly for this case.

3. Final Answer:

   minimize(lambda v: (v[0]-3)**2 + (v[1]+1)**2, [0, 0], bounds=[(None, None), (-2, 0)]) -> Option C

4. Quick Check:

   Bounds list matches variable order [OK]

Hint: Bounds list order matches variables order [OK]

Common Mistakes:

• Swapping bounds order for variables
• Using constraints instead of bounds incorrectly
• Setting equality constraint instead of bounds