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Optimality of Interval Reduction in the Fibonacci Search Method

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The Formal Theorem

Let f:[a,b]→R f: [a, b] \to \mathbb{R} be a unimodal function. Given a fixed number of function evaluations n n , the Fibonacci search algorithm achieves the minimum possible length of the final uncertainty interval Ln L_n such that Ln=b−aFn L_n = \frac{b-a}{F_n} , where Fn F_n is the n n -th Fibonacci number defined by F0=F1=1 F_0 = F_1 = 1 and Fn=Fn−1+Fn−2 F_n = F_{n-1} + F_{n-2} . This reduction is optimal for any zero-order derivative method, as it satisfies the optimality condition:
Ln=min⁡{x1,…,xn}(max⁡x∈[a,b]∣In(x)∣) L_n = \min_{\{x_1, \dots, x_n\}} \left( \max_{x \in [a, b]} |I_n(x)| \right)

Analytical Intuition.

Imagine you are an explorer searching for a hidden peak in a mountain range represented by a unimodal function. You only have n n days to investigate. Each day, you set up two observation posts to divide the current search space. The Fibonacci search is the 'master strategy' of efficiency. By choosing the distance between these posts according to the ratio of consecutive Fibonacci numbers, you ensure that every subsequent day, one of your previous posts becomes the new anchor for the following measurement. Unlike random or uniform sampling, the Fibonacci method is purely recursive; it wastes zero information from previous steps. As n n grows, the golden ratio ϕ≈1.618 \phi \approx 1.618 emerges naturally, dictating the most balanced split of the interval. By the n n -th step, you have squeezed the uncertainty interval to its theoretical physical limit. You aren't just searching; you are mathematically forcing the function to reveal its extremum with the absolute maximum speed allowed by the laws of information theory.
CAUTION

Institutional Warning.

Students frequently confuse Fibonacci search with the Golden Section search. While Golden Section search uses a constant ratio 1/ϕ 1/\phi and is a limiting case, Fibonacci search uses dynamic ratios tailored to the exact number of evaluations n n , providing a tighter interval for finite n n .

Academic Inquiries.

01

Why is Fibonacci search considered 'optimal'?

It is optimal because it minimizes the length of the final interval of uncertainty among all direct search methods that do not use derivatives, given a fixed number of function evaluations.

02

What happens if I don't know the exact value of n beforehand?

If n n is unknown, the Golden Section search is preferred as it maintains a constant ratio, whereas Fibonacci search requires n n to be defined at the start to determine the specific sequence of ratios.

Standardized References.

  • Definitive Institutional SourceBazaraa, M. S., Sherali, H. D., & Shetty, C. M., Nonlinear Programming: Theory and Algorithms.

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Optimality of Interval Reduction in the Fibonacci Search Method: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/fundamentals-of-optimization/optimality-of-interval-reduction-in-the-fibonacci-search-method

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