Minimizing scalar functions (minimize_scalar) in SciPy - Time & Space Complexity

Time Complexity: Minimizing scalar functions (minimize_scalar)

O(log n)

Understanding Time Complexity

When we use minimize_scalar from scipy, we want to find the lowest point of a simple function. Understanding how long this takes helps us plan for bigger problems.

We ask: How does the time to find the minimum grow as the function or search range changes?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

from scipy.optimize import minimize_scalar

def f(x):
    return (x - 2) ** 2 + 1

result = minimize_scalar(f, bounds=(0, 5), method='bounded')
print(result.x)
    

This code finds the minimum value of a simple function using Brent's method.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Evaluating the function f(x) multiple times.
• How many times: The optimizer calls f(x) repeatedly, narrowing down the minimum by testing points.

How Execution Grows With Input

The number of function checks grows slowly as the search narrows. Each step cuts down the search space efficiently.

Input Size (n)	Approx. Operations
10	About 5-10 function calls
100	About 7-15 function calls
1000	About 10-20 function calls

Pattern observation: The calls grow slowly, roughly proportional to the logarithm of the search range size.

Final Time Complexity

Time Complexity: O(log n)

This means the number of function checks grows slowly as the search range gets bigger, making the method efficient.

Common Mistake

[X] Wrong: "The optimizer checks every point in the range one by one."

[OK] Correct: The method smartly narrows the search space, so it does not check every point but focuses on promising areas.

Interview Connect

Knowing how optimization methods scale helps you explain your choices clearly and shows you understand efficient problem solving.

Self-Check

"What if we changed the method from 'brent' to a simple grid search? How would the time complexity change?"