Minimizing scalar functions (minimize_scalar) in SciPy - Mini Project: Build & Apply

Minimizing Scalar Functions with scipy.optimize.minimize_scalar

📖 Scenario: Imagine you are helping a small business find the best price to maximize their profit. The profit depends on the price, and you want to find the price that gives the lowest cost or highest profit loss. We will use a simple math function to represent this and find the minimum point.

🎯 Goal: You will create a function that calculates cost based on price, then use scipy.optimize.minimize_scalar to find the price that minimizes this cost.

📋 What You'll Learn

Create a function named cost_function that takes one argument x and returns (x - 3) ** 2 + 5.

Import minimize_scalar from scipy.optimize.

Use minimize_scalar to find the minimum of cost_function.

Store the result in a variable named result.

Print the minimum value found using result.fun and the location of the minimum using result.x.

💡 Why This Matters

🌍 Real World

Businesses often want to find the best price or quantity to minimize costs or maximize profits. Minimizing scalar functions helps find these optimal points quickly.

💼 Career

Data scientists and analysts use optimization techniques like <code>minimize_scalar</code> to solve real-world problems in finance, marketing, engineering, and more.

1

Create the cost function

Create a function called cost_function that takes one argument x and returns the value of (x - 3) ** 2 + 5.

SciPy

# Define the cost_function that calculates (x - 3) ** 2 + 5
# Your code here

Hint

Remember to use def to create a function and return the expression (x - 3) ** 2 + 5.

2

Import minimize_scalar

Import minimize_scalar from the scipy.optimize module.

SciPy

def cost_function(x):
    return (x - 3) ** 2 + 5

# Import minimize_scalar from scipy.optimize
# Your code here

Hint

Use the syntax from scipy.optimize import minimize_scalar to import the function.

3

Find the minimum using minimize_scalar

Use minimize_scalar with the cost_function to find the minimum. Store the result in a variable called result.

SciPy

from scipy.optimize import minimize_scalar

def cost_function(x):
    return (x - 3) ** 2 + 5

# Use minimize_scalar with cost_function and store in result
# Your code here

Hint

Call minimize_scalar with cost_function as the argument and assign it to result.

4

Print the minimum value and location

Print the minimum value found using result.fun and the location of the minimum using result.x. Use two separate print statements.

SciPy

from scipy.optimize import minimize_scalar

def cost_function(x):
    return (x - 3) ** 2 + 5

result = minimize_scalar(cost_function)

# Print the minimum value and the location of the minimum
# Your code here

Hint

Use print(result.fun) to show the minimum value and print(result.x) to show where it happens.

Practice

1. What does the minimize_scalar function in scipy do?

easy

A. Integrates a function over a range

B. Finds the maximum value of a function with multiple variables

C. Calculates the derivative of a function

D. Finds the minimum value of a function with one variable

Solution

1. Step 1: Understand the purpose of minimize_scalar

   The function is designed to find the lowest point (minimum) of a scalar function with one variable.

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

   Only Finds the minimum value of a function with one variable correctly describes this purpose; others describe different tasks.

3. Final Answer:

   Finds the minimum value of a function with one variable -> Option D

4. Quick Check:

   minimize_scalar finds minimum [OK]

Hint: Remember: minimize_scalar finds lowest point of one-variable function [OK]

Common Mistakes:

• Confusing minimize_scalar with maximization functions
• Thinking it works for multi-variable functions
• Mixing it up with derivative or integration functions

2. Which of the following is the correct way to import minimize_scalar from scipy.optimize?

easy

A. from scipy.optimize import minimize_scalar

B. import minimize_scalar from scipy.optimize

C. from scipy import minimize_scalar

D. import scipy.optimize.minimize_scalar

Solution

1. Step 1: Recall Python import syntax

   To import a specific function, use 'from module import function'.

2. Step 2: Match syntax to options

   from scipy.optimize import minimize_scalar matches correct syntax: 'from scipy.optimize import minimize_scalar'. Others are invalid Python syntax.

3. Final Answer:

   from scipy.optimize import minimize_scalar -> Option A

4. Quick Check:

   Correct import syntax = from scipy.optimize import minimize_scalar [OK]

Hint: Use 'from module import function' to import specific functions [OK]

Common Mistakes:

• Using 'import function from module' which is invalid
• Trying to import directly from 'scipy' without submodule
• Using dot notation in import statement incorrectly

3. What is the output of this code?

from scipy.optimize import minimize_scalar

result = minimize_scalar(lambda x: (x - 3)**2)
print(round(result.x, 2))

medium

A. 3.00

B. 0.00

C. -3.00

D. None

Solution

1. Step 1: Understand the function minimized

   The function is (x - 3)^2, which has its minimum at x = 3.

2. Step 2: Check the minimize_scalar result

   The optimizer finds x close to 3; rounding to 2 decimals gives 3.00.

3. Final Answer:

   3.00 -> Option A

4. Quick Check:

   Minimum of (x-3)^2 is at 3 [OK]

Hint: Minimum of (x - a)^2 is at x = a [OK]

Common Mistakes:

• Confusing minimum location with function value
• Not rounding result.x before printing
• Assuming minimum at zero without checking function

4. Identify the error in this code snippet:

from scipy.optimize import minimize_scalar

def f(x):
    return x**2

result = minimize_scalar(f, bounds=(1, 2))
print(result.x)

medium

A. minimize_scalar does not accept bounds argument

B. Function f should return negative values

C. Missing method='bounded' for bounded minimization

D. Bounds should be a list, not a tuple

Solution

1. Step 1: Check usage of bounds in minimize_scalar

   When using bounds, method='bounded' must be specified explicitly.

2. Step 2: Identify missing method argument

   The code misses method='bounded', so it will ignore bounds or error.

3. Final Answer:

   Missing method='bounded' for bounded minimization -> Option C

4. Quick Check:

   Bounds require method='bounded' [OK]

Hint: Use method='bounded' when specifying bounds [OK]

Common Mistakes:

• Forgetting method='bounded' with bounds
• Assuming bounds work without method argument
• Incorrect bounds format (tuple is correct though)

5. You want to find the minimum of f(x) = (x-1)^2 + 2 but only for x between 0 and 2. Which code correctly finds this minimum using minimize_scalar?

hard

A. minimize_scalar(lambda x: (x-1)**2 + 2, bounds=(0, 2))

B. minimize_scalar(lambda x: (x-1)**2 + 2, bounds=(0, 2), method='bounded')

C. minimize_scalar(lambda x: (x-1)**2 + 2, method='bounded')

D. minimize_scalar(lambda x: (x-1)**2 + 2, method='brent')

Solution

1. Step 1: Understand bounded minimization requirements

   To limit search to [0, 2], bounds must be set and method='bounded' used.

2. Step 2: Evaluate options for correct usage

   Only minimize_scalar(lambda x: (x-1)**2 + 2, bounds=(0, 2), method='bounded') uses both bounds and method='bounded' correctly.

3. Final Answer:

   minimize_scalar(lambda x: (x-1)**2 + 2, bounds=(0, 2), method='bounded') -> Option B

4. Quick Check:

   Bounds + method='bounded' needed for limited range [OK]

Hint: Use bounds with method='bounded' to limit search range [OK]

Common Mistakes:

• Omitting method='bounded' when using bounds
• Using method='brent' which ignores bounds
• Not specifying bounds when needed