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In ratio section method and algorithms for minimizing unimodal functions

This paper introduces a new ratio section method for minimizing unimodal functions that outperforms classical bisection, golden section, and modernized Brent algorithms by significantly reducing the number of function evaluations required through efficient recognition of monotone and flat-bottomed functions.

Original authors: Vladimir Kodnyanko

Published 2026-04-30

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Vladimir Kodnyanko

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to find the lowest point in a vast, foggy valley. You can't see the whole landscape at once; you can only stand at one spot, look around, and take a step. Your goal is to find the bottom of the valley (the minimum) using as few steps as possible. This is exactly what mathematicians do when they try to "minimize" a function.

This paper introduces a new, faster way to take those steps. Here is the breakdown of the author's ideas using simple analogies:

The Problem: The Old Ways of Searching

For a long time, mathematicians have used two main strategies to find that valley bottom:

The Bisection Method (The "Cut in Half" Approach): Imagine you have a long rope representing the valley. You cut it exactly in the middle, check the height, and then throw away the half that is higher. You repeat this, cutting the remaining rope in half every time. It's reliable, but it's a bit slow and rigid.
The Golden Section Search (The "Golden Ratio" Approach): This is a more sophisticated version of the first method. Instead of cutting the rope exactly in half, you cut it at a special "golden" spot (about 61.8% of the way). This is generally faster than cutting in half, but it still follows a strict, pre-set pattern.

The New Idea: The "Ratio Section" Method

The author, Vladimir Kodnyanko, proposes a new way to cut the rope. Instead of always cutting in half or at the golden ratio, he suggests cutting the rope at a customizable ratio.

Think of it like this: If you are walking down a hill, you don't always need to take a giant step or a tiny step. Sometimes, taking a step that is slightly closer to where you think the bottom is, rather than strictly following a rule, gets you there faster.

The paper introduces two versions of this new method:

1. The "Passive" Algorithm (RatioP)

This is the basic version. It's like a smart hiker who has a favorite step size.

How it works: It picks a spot based on a specific ratio (the author found that cutting the rope at about 20% of the way, rather than 50% or 61%, works best for most hills).
The Superpower: It has a special "eyesight" feature. If the valley is actually a flat plateau (a "flat bottom") or if the ground is just sloping steadily up or down (a "monotone" function), this method spots it immediately.
The Result: Because it can spot these special shapes quickly, it doesn't waste time taking unnecessary steps. In tests, it was 2.26 times faster than the old "cut in half" method and 1.72 times faster than the "golden ratio" method.

2. The "Active" Algorithm (RatioA)

This is the "super hiker." It doesn't just follow a ratio; it learns as it goes.

How it works: It uses the same smart ratio cutting as the passive version, but it also looks at the three most recent points it checked. If those three points look like they form a curve (a parabola), it uses a mathematical trick to guess the bottom of the curve instantly, rather than taking small steps.
The Result: This is the fastest method of all. It was 3.31 times faster than the "cut in half" method and 2.52 times faster than the golden ratio method.

The "Brent's Method" Upgrade

There is a famous, very fast method called Brent's Method that combines the reliability of the golden ratio with the speed of curve-guessing. The author took this famous method and swapped out its "golden ratio" step for his new "ratio section" step.

The Upgrade: This modernized version (called BrentM) became a beast. It was 1.69 times faster than the original Brent's method.
The Safety Net: The original Brent's method sometimes gets confused if the ground is perfectly flat or sloping straight up/down. The new version fixes this by instantly recognizing those shapes, so it never makes a mistake or gets stuck.

The Bottom Line

The paper tested these new methods against 20 different types of mathematical "hills" (some smooth, some flat, some jagged).

The Winner: The new Ratio Section methods are the fastest known ways to find the bottom of a single-variable valley.
Why it matters: In the world of computer optimization, "faster" means fewer calculations. Fewer calculations mean computers can solve complex problems in less time and with less energy.

In short, the author found a better way to slice the uncertainty interval (the "rope") that allows computers to find the lowest point of a curve much quicker than before, especially when the curve has flat spots or straight slopes.

1. Problem Statement

The paper addresses the challenge of minimizing unimodal functions of a single variable. While classical methods such as Bisection Search, Golden Section Search (GSS), and Brent's Method are well-established, they have limitations:

Bisection Search: Computationally redundant and generally slower.
Golden Section Search: Reliable but lacks mechanisms to accelerate convergence for specific function types (e.g., monotone or flat-bottomed functions).
General Efficiency: Existing methods often treat all unimodal functions uniformly, failing to exploit specific characteristics (like monotonicity or flat bottoms) that could allow for faster termination.

The goal is to develop a method that reduces the number of function evaluations required to find a minimum, particularly by adapting to the specific nature of the function being minimized.

2. Methodology

The author proposes a new Ratio Section Method and develops three specific algorithms based on it:

A. Classification of Functions

The methodology relies on classifying unimodal functions into three categories to apply specific optimization strategies:

Constant Functions: The entire domain is a flat bottom.
Monotone Functions: Strictly increasing or decreasing (possibly with a flat bottom on one side).
Strictly Unimodal Functions: Have a single minimum, potentially with a flat bottom region ( $\alpha < \beta$ ).

B. The Core Algorithms

RatioP (Passive Algorithm):
- Mechanism: Divides the uncertainty interval $[a, b]$ in a specific ratio $c$ . A new point $p$ is placed closer to the current minimum $m$ based on the ratio $c$ .
- Optimization: Unlike bisection (where $c=0.5$ ), the optimal ratio was found to be $c \approx 0.2$ . This places the new evaluation point closer to the current best estimate, compressing the uncertainty segment more aggressively.
- Special Handling: Includes a logic block to detect monotonicity and flat bottoms early. If three points with identical ordinates are found, the algorithm terminates immediately. If monotonicity is detected, the minimum is identified at the boundary.
RatioA (Active Algorithm):
- Mechanism: Builds upon RatioP but incorporates a Parabolic Interpolation step (a second-order method).
- Logic: Once three points are established that satisfy specific conditions (forming a "bracket" around a minimum), the algorithm calculates the vertex of the parabola passing through them to predict the minimum.
- Hybrid Approach: It uses RatioP to find the initial bracket and then switches to parabolic steps for faster convergence, falling back to ratio division if the parabola fails.
Modernized Brent's Method (BrentM):
- Modification: The classical Brent's method combines parabolic interpolation with the Golden Section Search. The author replaces the Golden Section component with the RatioP procedure.
- Enhancement: The modernized version also integrates the monotonicity and flat-bottom detection logic found in the RatioP algorithm, which the original Brent's method lacks.

3. Key Contributions

New Ratio Section Strategy: Introduction of a segment division method using a ratio $c$ (optimized at 0.2) rather than the fixed 0.5 (bisection) or the Golden Ratio (~0.618).
Early Termination Logic: Development of a specific algorithmic block (Mnt) to rapidly identify monotone functions and constant/flat-bottom functions, allowing the search to terminate in as few as 3–6 evaluations for these specific cases.
Optimization of Ratio $c$ : Empirical determination that $c=0.2$ yields the best performance for the RatioP algorithm across various function types.
Modernization of Brent's Method: Successfully integrating the ratio section and monotonicity detection into the standard Brent's algorithm, creating a hybrid that outperforms the original.

4. Experimental Results

The algorithms were tested on 20 unimodal functions of varying types (smooth, flat, monotone, piecewise) and compared against Bisection, Golden Section, and standard Brent's methods.

Performance Metrics: Measured by the total number of function evaluations ( $\Sigma k$ ).
RatioP vs. Classical Methods (All 20 functions):
- RatioP ( $c=0.2$ ) is 2.26 times faster than Bisection Search.
- RatioP is 1.72 times faster than Golden Section Search.
- For strictly unimodal functions (excluding monotone/flat cases), RatioP is still 1.40 times faster than GSS.
RatioA vs. RatioP:
- The active algorithm (RatioA) is significantly faster, reducing evaluations by an additional factor compared to the passive RatioP.
Modernized Brent (BrentM) vs. Classical Brent:
- BrentM is 1.69 times faster than the original Brent's method.
- BrentM is ~4 times faster than Bisection and ~3 times faster than GSS.
Robustness: The modernized Brent's method eliminates errors that occur in the original algorithm when minimizing monotone functions, providing exact results for this class.

5. Significance and Conclusion

The paper demonstrates that the Ratio Section Method is a superior alternative to traditional segment elimination techniques.

Efficiency: By optimizing the division ratio and adding logic to detect non-strictly unimodal functions, the method drastically reduces computational cost.
Versatility: The method handles a wider range of function behaviors (monotone, flat-bottom) more effectively than GSS or standard Brent's.
Practical Impact: The modernized Brent's method (BrentM) sets a new benchmark for 1D optimization, offering a combination of the reliability of segment elimination with the speed of parabolic interpolation and ratio-based acceleration.

The author concludes that the proposed methods, particularly the active RatioA and the modernized BrentM, represent the fastest known approaches for minimizing unimodal functions, outperforming established standards by significant margins (up to 3–4x in specific comparisons).