A Simplex Method for Function Minimization
The Computer Journal, 1965 · paper · Mathematics and statistics
Describes a method for minimizing a function of several variables by comparing values at the vertices of a simplex and replacing the worst vertex. The simplex adapts to the local landscape; the method is reported effective and compact, with a way to estimate the Hessian.
Checked: PDF from whale.am28.uni-tuebingen.de, fetched 5 Oct 2026. Page numbers are those of that PDF.
4 findings
Substance The text gives the expansion as "P** = γP* + (1 − γ)P̄" with γ defined as the ratio [P**P̄]/[P*P̄], but the Fig. 1 expansion box reads "Form P** = (1+γ)P* −γP̄", which corresponds to a distance ratio of 1+γ.
- p. 1 The method, expansion step
P** = yP* + (1 - y)P.
- p. 2 Fig. 1 flow diagram, expansion box
Form P** = (i+yJP* -TP
- p. 1 Method, expansion
P** = γP* + (1 − γ)P̄
- p. 2 Fig. 1 flow diagram
Form P** = (1+γ)P* −γP̄
Judge's note: Checked both crops against the page images. The text formula matches the stated definition of γ. The figure formula gives P** − P̄ = (1+γ)(P* − P̄), so the two definitions of the expansion step disagree. The text says Fig. 1 gives the complete method, so a reader implementing from the figure would expand by a different amount. No alternative parameterisation is stated anywhere in the paper.
Second check: another model agreed
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Substance Table 1, MEAN NUMBER, last strategy column (1, 1/4, 2): the entries 344, 335, 343, 253, 347 average 324.4, but the printed Mean is '322'.
- p. 3 Table 1, mean number, last strategy column
322
- p. 3 Table 1, mean number, last column
344 335 343 253 347 ... 322
- p. 3 Table 1, MEAN NUMBER block, last column, Mean row
344 335 343 253 347 322
- p. 3 Table 1, mean number, last column
344 335 343 253 347 ... Mean 322
Only the authors can say what is right here.
Judge's note: I read the table from the page image. Every other Mean in Table 1, in both the mean-number and minimum-number blocks, equals the simple average of its five rows after rounding. Only this column fails, by 2.4. From the paper alone I cannot tell whether an entry or the mean is misprinted.
Second check: another model agreed
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Substance With the quadratic written as "y = a0 + 2a'x + x'Bx", the matrix of second derivatives is 2B, yet the Appendix states "the information matrix is just B".
- p. 6 Appendix
and the information matrix is just B.
Only the authors can say what is right here.
Judge's note: Checked the coefficient formulas against this form. b_ii = 2(y_i + y0 − 2y0i) recovers B as the coefficient matrix, and x_min = −B⁻¹a is consistent with it. The Hessian of this form is therefore 2B, not B. The paper states no convention that absorbs the factor 2, and the later 2σ² multiplier follows the same convention, so only the authors can say which was intended.
Second check: another model agreed
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Reference or label The text gives the two extra function (2) strategies as 'α = 1, β = 1/3 or 1/4, γ = 3', but the Table 1 headers label them '(1, 1/3, 2)' and '(1, 1/4, 2)', i.e. γ = 2.
- p. 3 Results text, function (2) strategies
y = 3.
- p. 3 Table 1 strategy headers, last two columns
(1, h 2)
- p. 3 Results, function (2) trial
α = 1, β = ¼ or ¾, γ = 3.
- p. 3 Table 1 column headers
(1, ¼, 2) (1, ¾, 2)
Only the authors can say what is right here.
Judge's note: On the page 3 image the text reads β = 1/3 or 1/4, γ = 3. The Table 1 column headers read (1, 1/3, 2) and (1, 1/4, 2). The β values agree; only γ differs. Some reviewers read the β values as ¾ or ½, which is not what the image shows. The paper does not say which γ is correct.
Second check: another model agreed
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4 minor (typos and the like; most checked by the judge only)
Wording The text gives "144 evaluations for function (1)", but the 16 unasterisked Table 2 entries for function (1) average 143.2. Function (3) similarly averages 227.4 against the printed "228"; function (2) matches the printed 216.
- p. 3 Comparison with Powell's results
144 evaluations for function (1)
- p. 4 Table 2, function (1) column
130|
Only the authors can say what is right here.
Judge's note: Summed the Table 2 columns from the image. The gaps are small, and the means could have been computed from unrounded run data, or the 130† entry, which omits some arrangements, could weight differently. The discrepancy is therefore minor and needs the authors to clarify how the means were formed.
Second check: another model agreed
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Typo The reflection box in Fig. 1 reads "Form P = (1+α)P̄ - α Ph" and "Calculate y", without the asterisks on P* and y*; the next box tests "is y* < yl ?".
- p. 2 Fig. 1, first box
Form P = (1+α)P̄ - α Ph
Judge's note: The C001-b crop shows the stars are missing in the image itself, so this is not an extraction artefact. The star is used on the same quantities in the boxes that follow.
Second check: none (judge only)
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Typo Discussion, page 4: 'Out method is highly opportunist' should read 'Our method is highly opportunist'.
- p. 4 Discussion, second paragraph
Out method is highly opportunist
Judge's note: The page 4 image shows 'Out method' at the start of the second Discussion paragraph. This is a misprint for 'Our'.
Second check: none (judge only)
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Typo The Powell (1962) reference title is printed as "An iterative method for finding stationary value of a function of several variables", with the singular "value".
- p. 6 References, Powell (1962)
An iterative method for finding stationary value of a function of several variables
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: that Powell's 1962 Computer Journal title reads "stationary values" (plural). The C008-a crop confirms the singular form as printed; the true title cannot be checked from the files.
Second check: none (judge only)