Minimizing multivariate functions (minimize) in SciPy

Introduction

We use minimization to find the lowest point of a function with many variables. This helps us solve problems like finding the best settings or lowest cost.

Finding the best combination of ingredients to minimize cost in a recipe.

Adjusting parameters in a machine learning model to reduce error.

Optimizing investment portfolios to minimize risk.

Tuning settings in a manufacturing process to reduce waste.

Finding the shortest path or least energy in physics problems.

Syntax

SciPy

from scipy.optimize import minimize

result = minimize(fun, x0, method=None, jac=None, options=None)

# fun: function to minimize
# x0: starting guess (array-like)
# method: optimization algorithm (optional)
# jac: gradient function (optional)
# options: dictionary of solver options (optional)

The fun function should take an array and return a single number.

The x0 is your initial guess for the variables.

Examples

Minimize a simple function starting from (0,0).

SciPy

from scipy.optimize import minimize

def f(x):
    return (x[0]-1)**2 + (x[1]-2.5)**2

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

Use gradient information with BFGS method for faster convergence.

SciPy

from scipy.optimize import minimize

def f(x):
    return (x[0]-1)**2 + (x[1]-2.5)**2

def grad(x):
    return [2*(x[0]-1), 2*(x[1]-2.5)]

result = minimize(f, [0, 0], jac=grad, method='BFGS')
print(result.x)

Sample Program

This program finds the minimum of a simple function with two variables. It starts guessing at (0,0) and finds the point where the function is smallest.

SciPy

from scipy.optimize import minimize

def objective(x):
    # A function with two variables
    return (x[0] - 3)**2 + (x[1] + 1)**2 + 5

# Starting guess
x0 = [0, 0]

# Run minimization
result = minimize(objective, x0)

print('Minimum value found:', result.fun)
print('At position:', result.x)

Important Notes

Choosing a good starting point x0 can help the solver find the minimum faster.

Some methods need the gradient (jacobian) to work well, but it is optional.

The result object contains useful info like result.x (solution) and result.fun (minimum value).

Summary

Use scipy.optimize.minimize to find the lowest point of functions with many variables.

Provide the function and a starting guess to begin the search.

Check the result for the minimum value and where it happens.

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