Basin-hopping for global minima in SciPy - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is basin-hopping in optimization?

Basin-hopping is a method to find the lowest point (global minimum) of a function by jumping between different 'basins' or valleys in the function's landscape.

beginner

How does basin-hopping differ from simple local optimization?

Unlike local optimization that can get stuck in one valley, basin-hopping allows jumps to other valleys to explore more of the function and find the global minimum.

intermediate

What role does the 'acceptance test' play in basin-hopping?

The acceptance test decides if a new point is accepted based on its function value, allowing the algorithm to sometimes accept worse points to escape local minima.

beginner

Which Python library provides a basin-hopping implementation?

The scipy library offers a basin-hopping function in scipy.optimize to perform global optimization.

intermediate

What is a practical example where basin-hopping is useful?

Basin-hopping is useful in chemistry to find the most stable shape of a molecule by searching for the lowest energy arrangement.

What is the main goal of basin-hopping?

A Integrate a function numerically

B Find the maximum value of a function

C Calculate the derivative of a function

D Find the global minimum of a function

Basin-hopping is designed to find the global minimum by exploring multiple basins.

Which library in Python provides basin-hopping optimization?

A numpy

B scipy

C matplotlib

D pandas

The scipy.optimize module includes basin-hopping.

In basin-hopping, what is a 'basin'?

A A local minimum region in the function landscape

B A maximum point

C A random number generator

D A data visualization

A basin is a valley or local minimum area where the function value is low.

Why might basin-hopping accept a worse solution temporarily?

A To escape local minima and explore other basins

B Because it always chooses the worst option

C To speed up the calculation

D To reduce memory usage

Accepting worse solutions helps the algorithm jump out of local valleys to find the global minimum.

Which of these is NOT a step in basin-hopping?

A Random jump to a new point

B Local minimization from the new point

C Sorting data in ascending order

D Acceptance test to decide if new point is kept

Sorting data is unrelated to basin-hopping steps.

Explain how basin-hopping helps find a global minimum compared to local optimization.

Think about how jumping between valleys helps avoid getting stuck.

Describe a simple example of using scipy's basin-hopping to minimize a function.

Imagine you want to find the lowest point of a bumpy hill.

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