Basin-hopping for global minima in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of basin-hopping on a simple function

What is the output of the following code that uses basin-hopping to find the minimum of a quadratic function?

SciPy

import numpy as np
from scipy.optimize import basinhopping

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

initial_guess = [0, 0]
result = basinhopping(func, initial_guess, niter=10)
print(result.x.round(2))

A [nan, nan]

B [0.0, 0.0]

C [3.0, -1.0]

D [-3.0, 1.0]

💡 Hint

Think about where the quadratic function reaches its lowest value.

The function has a minimum at x=3 and y=-1. Basin-hopping finds this global minimum starting from [0,0].

❓ data_output

intermediate

Number of iterations performed

After running basin-hopping with niter=15 on the function below, how many iterations does the result report?

SciPy

import numpy as np
from scipy.optimize import basinhopping

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

result = basinhopping(func, [0, 0], niter=15)
print(result.nit)

A 15

B 14

C 16

D 0

💡 Hint

The nit attribute shows the number of iterations performed.

The niter parameter sets the number of basin-hopping iterations. The result.nit matches this value.

🧠 Conceptual

advanced

Understanding the role of the 'stepsize' parameter

What effect does increasing the 'stepsize' parameter have in the basin-hopping algorithm?

A It decreases the number of iterations performed.

B It increases the size of random jumps, helping escape local minima more easily.

C It changes the function to be minimized.

D It sets the tolerance for convergence.

💡 Hint

Think about how the algorithm explores the search space.

The 'stepsize' controls how far the algorithm jumps randomly to explore new areas, helping to avoid getting stuck.

🔧 Debug

advanced

Identify the error in basin-hopping usage

What error will this code raise when executed?

SciPy

from scipy.optimize import basinhopping

def f(x):
    return x**2

result = basinhopping(f, 5, niter='10')

A No error, runs successfully

B ValueError: initial guess must be an array

C SyntaxError: invalid syntax

D TypeError: 'str' object cannot be interpreted as an integer

💡 Hint

Check the type of the niter parameter.

The niter parameter must be an integer, but a string was passed, causing a TypeError.

🚀 Application

expert

Choosing the best initial guess for basin-hopping

You want to find the global minimum of a function with many local minima. Which initial guess strategy is best to improve basin-hopping results?

A Use multiple random initial guesses and run basin-hopping separately for each.

B Use a fixed initial guess at zero always.

C Use the derivative of the function as the initial guess.

D Use the maximum value of the function as the initial guess.

💡 Hint

Think about how to avoid getting stuck in local minima.

Starting from multiple points increases the chance of finding the global minimum in complex landscapes.

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