Basin-hopping for global minima in SciPy

Introduction

Basin-hopping helps find the lowest point (global minimum) of a complex function. It avoids getting stuck in small dips (local minima) by jumping around.

When you want to find the best solution in a problem with many ups and downs.

When simple methods get stuck in a local low point and miss the true lowest point.

When optimizing functions that are not smooth or have many peaks and valleys.

When you want a reliable way to explore different areas of the solution space.

When you have a function where guessing the starting point is hard.

Syntax

SciPy

from scipy.optimize import basinhopping

result = basinhopping(func, x0, niter=100, stepsize=0.5)

func is the function you want to minimize.

x0 is the starting guess for the solution.

Examples

Finds the minimum of a simple parabola starting from 0.

SciPy

from scipy.optimize import basinhopping

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

result = basinhopping(func, 0)
print(result.x)

Minimizes a function with two variables starting from [1,1].

SciPy

from scipy.optimize import basinhopping

def func(x):
    return x[0]**2 + x[1]**2 + 10

result = basinhopping(func, [1, 1], niter=50, stepsize=1)
print(result.x)

Sample Program

This code tries to find the lowest point of a wavy function that has many dips. Basin-hopping jumps around to avoid getting stuck in small dips and finds the lowest one.

SciPy

from scipy.optimize import basinhopping
import numpy as np

def func(x):
    # A function with many local minima
    return np.sin(3 * x) + (x - 1)**2

# Start at x=0
result = basinhopping(func, 0, niter=100, stepsize=0.5)

print(f"Global minimum found at x = {result.x:.4f}")
print(f"Function value at minimum = {result.fun:.4f}")

Important Notes

Basin-hopping combines random jumps with local minimization to explore the function better.

You can adjust niter (number of jumps) and stepsize (jump size) to improve results.

It works well for functions with many local minima but can be slower than simple methods.

Summary

Basin-hopping helps find the lowest point in tricky functions with many dips.

It uses random jumps plus local searches to avoid getting stuck.

You can control how many jumps and how big they are to balance speed and accuracy.

Practice

1. What is the main purpose of the basin-hopping algorithm in optimization?

easy

A. To calculate the derivative of a function

B. To perform a simple linear regression

C. To sort a list of numbers efficiently

D. To find the global minimum of a function with many local minima

Solution

1. Step 1: Understand the goal of basin-hopping

   Basin-hopping is designed to find the lowest point (global minimum) in complex functions that have many dips (local minima).

2. Step 2: Compare with other options

   Options A, B, and C describe unrelated tasks: differentiation, regression, and sorting, which are not the purpose of basin-hopping.

3. Final Answer:

   To find the global minimum of a function with many local minima -> Option D

4. Quick Check:

   Basin-hopping = global minimum search [OK]

Hint: Basin-hopping = global minimum finder in tricky functions [OK]

Common Mistakes:

• Confusing basin-hopping with simple optimization methods
• Thinking it sorts or differentiates functions
• Assuming it only finds local minima

2. Which of the following is the correct way to import the basin-hopping function from scipy?

easy

A. from scipy import basinhopping

B. import scipy.basinhopping

C. from scipy.optimize import basinhopping

D. import basinhopping from scipy.optimize

Solution

1. Step 1: Recall correct import syntax in Python

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

2. Step 2: Match with scipy.optimize and basinhopping

   The basin-hopping function is inside scipy.optimize, so the correct import is 'from scipy.optimize import basinhopping'.

3. Final Answer:

   from scipy.optimize import basinhopping -> Option C

4. Quick Check:

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

Hint: Use 'from scipy.optimize import basinhopping' to import [OK]

Common Mistakes:

• Using incorrect import order or syntax
• Trying to import basin-hopping directly from scipy
• Using 'import basinhopping from ...' which is invalid

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

import numpy as np
from scipy.optimize import basinhopping

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

result = basinhopping(func, x0=0, niter=5)
print(round(result.fun, 2))

medium

A. 5.00

B. 0.00

C. 9.00

D. 3.00

Solution

1. Step 1: Understand the function and its minimum

   The function is (x - 3)^2 + 5, which has its minimum value at x=3, and the minimum value is 5.

2. Step 2: Analyze basin-hopping output

   Basin-hopping tries to find the global minimum. Starting at x0=0, after 5 iterations, it should find near x=3, so the function value is near 5.

3. Final Answer:

   5.00 -> Option A

4. Quick Check:

   Minimum value of (x-3)^2+5 = 5 [OK]

Hint: Minimum of (x-3)^2+5 is 5 at x=3 [OK]

Common Mistakes:

• Confusing minimum value with x-coordinate
• Assuming starting point is the minimum
• Ignoring the constant +5 in the function

4. Identify the error in the following code using basin-hopping:

from scipy.optimize import basinhopping

def f(x):
    return x**2

result = basinhopping(f, x0=[1, 2], niter=10)
print(result.x)

medium

A. No error; code runs correctly

B. Function f must return a scalar, but it returns a list

C. x0 should be a numpy array, not a list

D. x0 should be a scalar, not a list

Solution

1. Step 1: Check input types for basin-hopping

   basinhopping accepts x0 as a scalar or array-like. A list like [1, 2] is valid and converted to numpy array internally.

2. Step 2: Verify function output

   Function f(x) = x**2. For vector x = np.array([1,2]), it returns np.array([1,4]), not a scalar. Optimization requires scalar objective function value.

3. Step 3: Test code behavior

   The code raises an error because the objective function returns an array instead of scalar.

4. Final Answer:

   Function f must return a scalar, but it returns a list -> Option B

5. Quick Check:

   Objective func must return scalar [OK]

Hint: basinhopping objective must return scalar for vector x0 [OK]

Common Mistakes:

• Assuming no error; overlooking non-scalar function return
• Thinking x0 list causes the error
• Believing x0 must be scalar or explicit numpy array

5. You want to find the global minimum of a function with many local minima using basin-hopping. Which parameter should you adjust to increase the chance of escaping local minima?

hard

A. Increase the 'stepsize' parameter to allow bigger jumps

B. Decrease the 'niter' parameter to reduce iterations

C. Set 'minimizer_kwargs' to None

D. Use a fixed starting point without randomization

Solution

1. Step 1: Understand basin-hopping parameters

   'stepsize' controls how big the random jumps are between local minimizations. Bigger steps help jump out of local minima.

2. Step 2: Evaluate options

   Decreasing 'niter' reduces attempts, lowering success. Setting 'minimizer_kwargs' to None disables local minimization, which is needed. Fixed start without randomization limits exploration.

3. Final Answer:

   Increase the 'stepsize' parameter to allow bigger jumps -> Option A

4. Quick Check:

   Bigger stepsize = better escape from local minima [OK]

Hint: Bigger stepsize helps jump out of local minima [OK]

Common Mistakes:

• Reducing iterations thinking it speeds up convergence
• Disabling local minimization by setting minimizer_kwargs to None
• Using fixed start point limits search space