Basin-hopping for global minima in SciPy - Time & Space Complexity

Time Complexity: Basin-hopping for global minima

O(n x m)

Understanding Time Complexity

We want to understand how the time needed to find the lowest point using basin-hopping changes as the problem size grows.

How does the number of steps and calculations grow when we try to find the global minimum?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

from scipy.optimize import basinhopping

def func(x):
    return x**4 - 3*x**3 + 2

result = basinhopping(func, x0=[0], niter=100)
print(result.x, result.fun)

This code tries to find the lowest value of a function by jumping around and checking many points.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Running local minimizations repeatedly after random jumps.
• How many times: The local minimization runs once per iteration, here 100 times.

How Execution Grows With Input

Each iteration runs a local search that depends on the problem size, and the total steps grow with the number of iterations.

Input Size (n)	Approx. Operations
10	About 100 local searches, each with small cost
100	100 local searches, each more costly due to bigger input
1000	100 local searches, each much more costly as input grows

Pattern observation: The total time grows roughly with the number of iterations times the cost of each local search, which grows with input size.

Final Time Complexity

Time Complexity: O(n \times m)

This means the time grows with the number of iterations m times the cost of local minimization on input size n.

Common Mistake

[X] Wrong: "The time only depends on the number of iterations, not the input size."

[OK] Correct: Each local search inside an iteration depends on input size, so bigger problems take longer per step.

Interview Connect

Understanding how iterative optimization methods scale helps you explain algorithm choices clearly and confidently.

Self-Check

"What if we increase the number of iterations instead of input size? How would the time complexity change?"

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