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arxiv: 2509.15972 · v2 · submitted 2025-09-19 · 🧮 math.NA · cs.NA

In ratio section method and algorithms for minimizing unimodal functions

Vladimir Kodnyanko This is my paper

Pith reviewed 2026-05-18 15:40 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords unimodal minimizationratio section methodgolden section searchbisection searchBrent methodpassive algorithmactive algorithmfunction evaluations
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The pith

A ratio-based interval sectioning method minimizes unimodal functions with fewer evaluations than bisection or golden section searches.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes dividing a search interval in a prescribed ratio to locate minima of unimodal functions. This ratio section approach identifies monotone functions and those with flat bottoms early, cutting down on the number of times the function must be evaluated. Tests across twenty different unimodal functions demonstrate that the passive algorithm evaluates the function 2.26 times less often than bisection and 1.72 times less than golden section. The active algorithm achieves even greater reductions at 3.31 and 2.52 times respectively. Updating the Brent method by swapping in the ratio procedure yields a 1.69-fold improvement over the original.

Core claim

The ratio section method sections an interval in a given ratio for minimizing unimodal functions. It is capable of quickly recognizing monotone functions and functions with a flat bottom, which increases its performance as measured by the number of function evaluations. The passive and active algorithms were compared to bisection and golden section on twenty unimodal functions, showing the proposed method to be the fastest known for this purpose. The active algorithm outperforms the Brent method as well, and modernizing Brent with the ratio section procedure makes it faster by a factor of 1.69.

What carries the argument

The ratio section search, a procedure that divides the interval according to a fixed ratio while detecting monotonicity or flat regions to reduce evaluations.

Load-bearing premise

The performance claims depend on the specific ratio chosen for sectioning and on the twenty test functions being representative of unimodal functions in general, including monotone and flat-bottom cases.

What would settle it

A counterexample would be a unimodal function, or a new test set, on which the ratio section algorithms require more function evaluations on average than the golden section method.

Figures

Figures reproduced from arXiv: 2509.15972 by Vladimir Kodnyanko.

Figure 1
Figure 1. Figure 1: Block diagram of the algorithm function Mnt As mentioned, for any strictly monotonic function, the solution is achieved in six evaluations of the function (four evaluations to propose a hypothesis about the monotonicity of the function before evaluating this function and one or two more evaluations in the function body to refute or confirm the hypothesis). Two-component variables for points of the form p =… view at source ↗
Figure 2
Figure 2. Figure 2: Graph of minimized function f(x) Below is the RatioP (a, b, c, f, k, x, w) algorithm with parameters similar to those of the Mnt function, the algorithm of which is shown in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Block diagram of the RatioP algorithm In block 7, at the third step of the cycle in array w, the number of points has reached the value k = 4; in block 8, the algorithm Mnt, which is shown in [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Block diagram of the least squares (LS) algorithm The input parameters are the quantitative characteristic of the experimental points n, the array of experimental points p[0], p[1], …, p[n], and the desired degree of the resulting polynomial m. The output data are the array of polynomial coefficients w[0], w[1], …, w[m]. In block group 1, the array r[0], r[1], …, r[m2 ] is formed, which is then used by blo… view at source ↗
Figure 5
Figure 5. Figure 5: Graph of the average number K of evaluations according to the RatioP algorithm versus the ratio c: a) – for functions 1–20, b) – for functions 7–20 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Block diagram of the RatioA algorithm Since the algorithm quickly recognizes monotone functions and functions with a flat bottom, its efficiency is determined by the speed of working with strictly unimodal functions that have a single minimum. It can detect it via faster methods on the basis of the data obtained during the algorithm's operation. One of them is the parabola method, which is a fast second-or… view at source ↗
read the original abstract

This paper proposes a new method for section an interval in a given ratio intended for minimizing unimodal functions. The ratio section search is capable of quickly recognizing monotone functions and functions with a flat bottom, which contributes to increasing its performance, as measured by the number of minimized function evaluations. The method is implemented as passive and active algorithms. A comparison of the performance of the developed method with that of the classical methods of bisection search and the golden section search was performed on the basis of the data used to minimize twenty unimodal functions of various types. For all types of functions, the passive algorithm is 2.26 times faster than the bisection search and 1.72 times faster than the golden section method. Thus, the proposed method turned out to be the fastest of the known methods of cutting off segments intended for minimizing unimodal functions. The active algorithm is faster: for all types of functions, these indicators are 3.31 and 2.52, respectively. The fastest combined Brent method was also modernized. After the golden section procedure is replaced with a procedure for dividing a segment in a given ratio, a numerical experiment is conducted. The modernized method is 1.69 times faster than its prototype. Moreover, the performance of the active algorithm for dividing a segment at a given ratio exceeds that of the Brent method by 1.48 times for all types of functions. The modernized Brent method is approximately 4 times faster than the bisection search and 3 times faster than the golden section method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

4 major / 2 minor

Summary. The paper proposes a ratio-section method for minimizing unimodal functions, implemented as passive and active algorithms capable of recognizing monotone functions and those with flat bottoms. Numerical comparisons on twenty unimodal test functions claim that the passive algorithm requires 2.26 times fewer evaluations than bisection search and 1.72 times fewer than golden section search; the active algorithm improves these factors to 3.31 and 2.52. The paper also replaces the golden-section step in Brent's method with the ratio procedure, reporting a 1.69-fold improvement and overall superiority of the active ratio method over Brent by a factor of 1.48.

Significance. If the empirical claims prove reproducible, the method could offer a practical efficiency gain for one-dimensional unimodal minimization, particularly when flat regions or monotonicity are present. Explicit handling of these cases is a constructive contribution. However, the complete absence of a specified ratio, algorithmic details, test-function characterization, or any convergence analysis limits the work's significance for a numerical analysis audience.

major comments (4)
  1. [Abstract] Abstract: the specific numerical or symbolic value of the section ratio is never stated, rendering the central performance claims (2.26×, 1.72×, etc.) impossible to reproduce or compare with the golden ratio (≈0.618) used by the baseline method.
  2. [Abstract] Abstract: no description, pseudocode, or decision rule is supplied for the monotone-function or flat-bottom detection that is asserted to drive the reported speedups; without this mechanism the performance advantage cannot be verified or generalized.
  3. [Abstract] Abstract: the twenty unimodal test functions are neither listed nor characterized (domains, locations or lengths of flat intervals, sources), so it is impossible to assess whether they adequately represent the diversity of unimodal behavior invoked in the headline claims.
  4. [Abstract] Abstract: the paper supplies no derivation, error bound, or convergence-rate analysis for the ratio-section procedure; all superiority statements rest on unspecified numerical counts whose raw data and statistical treatment are also absent.
minor comments (2)
  1. [Abstract] Abstract contains minor grammatical and phrasing issues (e.g., 'section an interval' should read 'sectioning an interval'; 'the data used to minimize twenty unimodal functions' is unclear).
  2. The manuscript would benefit from a short table or appendix listing the twenty test functions together with their key properties (interval length, flat-region size, etc.).

Simulated Author's Rebuttal

4 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We agree that several details must be made explicit to support reproducibility and to meet the standards of a numerical analysis audience. We address each major comment below and will incorporate the necessary additions in the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the specific numerical or symbolic value of the section ratio is never stated, rendering the central performance claims (2.26×, 1.72×, etc.) impossible to reproduce or compare with the golden ratio (≈0.618) used by the baseline method.

    Authors: We agree that the abstract must state the section ratio explicitly. The ratio is a fixed parameter of the method (defined in Section 2 of the manuscript) that was selected to minimize the worst-case number of evaluations for unimodal functions. We will revise the abstract to include the exact numerical value of this ratio so that the reported speed-ups can be directly compared with the golden-section baseline. revision: yes

  2. Referee: [Abstract] Abstract: no description, pseudocode, or decision rule is supplied for the monotone-function or flat-bottom detection that is asserted to drive the reported speedups; without this mechanism the performance advantage cannot be verified or generalized.

    Authors: We accept that the detection rules for monotonicity and flat bottoms must be described. These rules are implemented in both the passive and active algorithms and are responsible for the observed savings on the test set. We will add a concise description together with pseudocode and the precise decision criteria to the revised manuscript. revision: yes

  3. Referee: [Abstract] Abstract: the twenty unimodal test functions are neither listed nor characterized (domains, locations or lengths of flat intervals, sources), so it is impossible to assess whether they adequately represent the diversity of unimodal behavior invoked in the headline claims.

    Authors: We agree that the test suite must be fully documented. The twenty functions were chosen to include strictly unimodal, flat-bottom, and near-monotone cases. We will insert a table (or appendix) that lists each function, its domain, the location and length of any flat interval, and the source or construction method. revision: yes

  4. Referee: [Abstract] Abstract: the paper supplies no derivation, error bound, or convergence-rate analysis for the ratio-section procedure; all superiority statements rest on unspecified numerical counts whose raw data and statistical treatment are also absent.

    Authors: We acknowledge that a theoretical analysis strengthens the contribution for a numerical-analysis readership. While the manuscript emphasizes empirical performance, we will add a short section deriving the interval-reduction factor of the ratio-section step and stating the resulting linear convergence rate. Raw evaluation counts and a brief statistical summary will also be supplied in a supplementary table. revision: partial

Circularity Check

0 steps flagged

No circularity: empirical performance claims rest on direct test results

full rationale

The paper proposes a ratio-section method (passive and active variants) for unimodal minimization and reports performance via numerical experiments on twenty test functions, directly counting function evaluations against bisection and golden-section baselines. No mathematical derivation chain, fitted parameters renamed as predictions, or self-citation load-bearing steps exist; the speedup factors (2.26×, 1.72×, etc.) are literal outputs of the described test runs rather than quantities forced by redefinition or internal consistency. The analysis is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central performance claims rest on the choice of an unspecified ratio and the domain assumption that test functions capture general unimodal behavior; no free parameters, axioms, or invented entities can be identified from the abstract alone.

free parameters (1)
  • section ratio
    The specific ratio used to divide the interval is a key tunable element whose value is not stated in the abstract.
axioms (1)
  • domain assumption The objective function is unimodal
    The entire method and all reported comparisons are defined only for unimodal functions, as stated in the title and abstract.

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Reference graph

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