Global optimization (differential_evolution) in SciPy

Introduction

Global optimization helps find the best solution when many possible answers exist. Differential evolution is a method that tries many solutions and improves them step-by-step to find the best one.

When you want to find the lowest or highest value of a complex function with many peaks and valleys.

When simple methods get stuck in a bad solution and can't find the best answer.

When you have limits on the values your solution can take, like a range for each variable.

When you want a method that works well even if the function is not smooth or has many ups and downs.

Syntax

SciPy

from scipy.optimize import differential_evolution

result = differential_evolution(func, bounds, strategy='best1bin', maxiter=1000, popsize=15, tol=0.01)

# func: function to minimize
# bounds: list of (min, max) for each variable
# result.x: best solution found
# result.fun: function value at best solution

func must take a list or array of variables and return a single number.

bounds define the search space for each variable.

Examples

Minimize a simple function with two variables between -5 and 5.

SciPy

from scipy.optimize import differential_evolution

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

bounds = [(-5, 5), (-5, 5)]
result = differential_evolution(func, bounds)
print(result.x, result.fun)

Minimize a shifted parabola with custom iteration and population size.

SciPy

from scipy.optimize import differential_evolution

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

bounds = [(-3, 3), (-3, 3)]
result = differential_evolution(func, bounds, maxiter=500, popsize=10)
print(result.x, result.fun)

Sample Program

This program finds the minimum of the Rosenbrock function, a common test problem in optimization. It searches within the given bounds and prints the best solution and its function value.

SciPy

from scipy.optimize import differential_evolution

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

bounds = [(-2, 2), (-1, 3)]

result = differential_evolution(rosenbrock, bounds)

print(f"Best solution: {result.x}")
print(f"Function value at best solution: {result.fun}")

Important Notes

Differential evolution works well for many problems but can be slower than simple methods for easy problems.

Adjusting maxiter and popsize changes how long the search runs and how thoroughly it explores.

Always check the result to see if the solution makes sense for your problem.

Summary

Differential evolution is a global optimization method that tries many solutions and improves them.

It works well for complex functions with many variables and tricky shapes.

You provide the function and bounds, and it returns the best solution found.

Practice

1. What is the main purpose of the differential_evolution function in scipy.optimize?

easy

A. To perform linear regression on data

B. To calculate the derivative of a function

C. To find the global minimum of a function within given bounds

D. To generate random numbers for simulations

Solution

1. Step 1: Understand the purpose of differential evolution

   Differential evolution is a global optimization algorithm designed to find the minimum value of a function over a specified range.

2. Step 2: Compare options with the purpose

   Only To find the global minimum of a function within given bounds describes finding the global minimum within bounds, which matches the function's purpose.

3. Final Answer:

   To find the global minimum of a function within given bounds -> Option C

4. Quick Check:

   Global optimization = find minimum [OK]

Hint: Remember: differential_evolution finds global minimum [OK]

Common Mistakes:

• Confusing optimization with differentiation
• Thinking it performs regression
• Assuming it generates random numbers only

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

easy

A. import differential_evolution from scipy.optimize

B. from scipy.optimize import differential_evolution

C. from scipy import differential_evolution.optimize

D. import scipy.optimize.differential_evolution

Solution

1. Step 1: Recall Python import syntax

   The correct syntax to import a function from a module is: from module import function.

2. Step 2: Match syntax with options

   from scipy.optimize import differential_evolution matches the correct syntax exactly. Other options have incorrect order or syntax.

3. Final Answer:

   from scipy.optimize import differential_evolution -> Option B

4. Quick Check:

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

Hint: Use 'from module import function' for importing functions [OK]

Common Mistakes:

• Using 'import function from module' which is invalid
• Trying to import submodules incorrectly
• Using dot notation in import statements wrongly

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

from scipy.optimize import differential_evolution

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

result = differential_evolution(func, bounds=[(-5, 5), (-5, 5)])
print(round(result.fun, 2))

medium

A. 0.00

B. 5.00

C. 13.00

D. Error due to wrong bounds

Solution

1. Step 1: Understand the function and bounds

   The function calculates the sum of squares of (x0-2) and (x1+3). The minimum is at x0=2, x1=-3, which is inside the bounds (-5,5).

2. Step 2: Evaluate the minimum function value

   At x0=2 and x1=-3, func returns (2-2)^2 + (-3+3)^2 = 0 + 0 = 0. So the minimum function value is 0.

3. Final Answer:

   0.00 -> Option A

4. Quick Check:

   Minimum value at (2,-3) = 0 [OK]

Hint: Minimum of squared terms is zero at exact target points [OK]

Common Mistakes:

• Misunderstanding bounds causing error
• Calculating function value at wrong points
• Ignoring rounding in output

4. Identify the error in the following code using differential_evolution and how to fix it:

from scipy.optimize import differential_evolution

def f(x):
    return x**2

bounds = [(-3, 3)]
result = differential_evolution(f, bounds)
print(result.x)

medium

A. No error; code runs correctly

B. Bounds should be a tuple, not a list; fix by changing bounds to a tuple

C. Missing import of numpy; fix by adding import numpy as np

D. Function f expects a scalar but receives an array; fix by defining f(x) to accept array input

Solution

1. Step 1: Analyze function input requirements

   The differential_evolution function passes an array to f, but f expects a scalar and tries to square it directly, which causes an error.

2. Step 2: Fix function to accept array input

   Modify f to accept an array input and return a scalar, for example: def f(x): return x[0]**2.

3. Final Answer:

   Function f expects a scalar but receives an array; fix by defining f(x) to accept array input -> Option D

4. Quick Check:

   Function input shape mismatch = Function f expects a scalar but receives an array; fix by defining f(x) to accept array input [OK]

Hint: Functions must accept array input for differential_evolution [OK]

Common Mistakes:

• Assuming function gets scalar input
• Incorrect bounds format
• Ignoring error messages about input types

5. You want to minimize the function f(x) = sin(x[0]) + cos(x[1]) over the bounds x[0] in [0, 6] and x[1] in [-3, 3]. Which of the following code snippets correctly uses differential_evolution to find the minimum and print the best input values?

hard

A. from scipy.optimize import differential_evolution
import numpy as np

def f(x):
return np.sin(x[0]) + np.cos(x[1])

bounds = [(0, 6), (-3, 3)]
result = differential_evolution(f, bounds)
print(result.x)

B. from scipy.optimize import differential_evolution
import numpy as np

def f(x):
return np.sin(x) + np.cos(x)

bounds = [(0, 6), (-3, 3)]
result = differential_evolution(f, bounds)
print(result.x)

C. from scipy.optimize import differential_evolution
import numpy as np

def f(x):
return np.sin(x[0]) + np.cos(x[1])

bounds = [0, 6, -3, 3]
result = differential_evolution(f, bounds)
print(result.x)

D. from scipy.optimize import differential_evolution
import numpy as np

def f(x):
return np.sin(x[0]) + np.cos(x[1])

bounds = [(0, 6), (-3, 3)]
result = differential_evolution(f)
print(result.x)

Solution

1. Step 1: Check function definition

   The function f correctly uses array indexing for x[0] and x[1] and applies numpy sine and cosine functions.

2. Step 2: Verify bounds and function call

   Bounds are correctly given as a list of tuples for each variable. The differential_evolution call includes both the function and bounds.

3. Final Answer:

   Code snippet C correctly defines the function, bounds, and calls differential_evolution -> Option A

4. Quick Check:

   Correct function and bounds usage = from scipy.optimize import differential_evolution import numpy as np def f(x): return np.sin(x[0]) + np.cos(x[1]) bounds = [(0, 6), (-3, 3)] result = differential_evolution(f, bounds) print(result.x) [OK]

Hint: Use list of tuples for bounds and array input function [OK]

Common Mistakes:

• Passing bounds as flat list instead of list of tuples
• Defining function without indexing array input
• Calling differential_evolution without bounds argument