Minimizing multivariate functions (minimize) in SciPy - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to import the minimize function from scipy.optimize.

SciPy

from scipy.optimize import [1]

Drag options to blanks, or click blank then click option'

Amin

Boptimize

Cminimize

Dminimizer

Attempts:

3 left

💡 Hint

Common Mistakes

Using 'min' instead of 'minimize'.

Trying to import 'optimize' instead of 'minimize'.

The minimize function is imported directly from scipy.optimize to perform minimization.

2fill in blank

medium

Complete the code to define a function f(x) that returns x[0] squared plus x[1] squared.

SciPy

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

Drag options to blanks, or click blank then click option'

A*

B**

C+

D-

Attempts:

3 left

💡 Hint

Common Mistakes

Using '+' instead of '*' for squaring.

Using '**' inconsistently in the expression.

To square a number, you multiply it by itself using *. So x[0] * x[0] is correct.

3fill in blank

hard

Fix the error in the code to minimize function f starting from initial guess [1, 1].

SciPy

result = minimize(f, [1])

Drag options to blanks, or click blank then click option'

A[1, 1]

B1, 1

C(1, 1)

D{1, 1}

Attempts:

3 left

💡 Hint

Common Mistakes

Passing separate numbers instead of a list.

Using a set instead of a list.

The initial guess must be a list or array. So [1, 1] is correct.

4fill in blank

hard

Fill both blanks to create a dictionary of squares for numbers 1 to 5, but only include numbers greater than 3.

SciPy

squares = {x: x[1]2 for x in range(1, 6) if x [2] 3}

Drag options to blanks, or click blank then click option'

A**

B%

C>

D+

Attempts:

3 left

💡 Hint

Common Mistakes

Using '+' instead of '**' for squaring.

Using '<' instead of '>' in the condition.

The expression x**2 squares x. The condition x > 3 filters numbers greater than 3.

5fill in blank

hard

Fill all three blanks to create a dictionary with uppercase keys and values only if value is positive.

SciPy

result = [1]: [2] for k, v in data.items() if v [3] 0}

Drag options to blanks, or click blank then click option'

Ak.upper()

Bv

C>

Dk.lower()

Attempts:

3 left

💡 Hint

Common Mistakes

Using k.lower() instead of k.upper().

Using '<' instead of '>' in the condition.

k.upper() converts keys to uppercase, v is the value, and v > 0 filters positive values.

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