Implemented mechanisms

This page lists the implemented primitive functions, transforms, value transforms, and composition modes. YAML/spec parsers normalize case, spaces, hyphens, and underscores before matching names.

Base functions

Primitive functions are exposed through BasicFunctionId. Numeric IDs and names can both be used in YAML.

ID	Name	Notes
1	Sphere	Unimodal, smooth, separable.
2	Ellipsoidal	Unimodal, high-conditioned.
3	SumDifferentPowers	Unimodal, variable powers.
4	BuecheRastrigin	Multimodal Rastrigin-family landscape.
5	LinearSlope	Unimodal sloped landscape.
6	AttractiveSector	Unimodal, asymmetric sector structure.
7	StepEllipsoidal	Unimodal stepped high-conditioned landscape.
8	StepRastrigin	Multimodal stepped Rastrigin-family landscape.
9	Rosenbrock	Unimodal valley, nonseparable.
10	Ackley	Multimodal.
11	Rastrigin	Multimodal, separable in primitive coordinates.
12	Griewank	Multimodal.
13	Schwefel	Multimodal.
14	SharpRidge	Unimodal ridge.
15	Weierstrass	Multimodal, rugged.
16	SchafferF7	Multimodal.
17	SchafferF7Cond1000	Multimodal, conditioned Schaffer variant.
18	GriewankRosenbrock	Multimodal hybrid.
19	Gallagher21	Multimodal peaks.
20	Katsuura	Multimodal.
21	LunacekBiRastrigin	Multimodal double-funnel Rastrigin-family landscape.
22	Zakharov	Unimodal.
23	Levy	Multimodal.
24	Michalewicz	Multimodal.
25	DixonPrice	Unimodal.
26	BentCigar	Unimodal, high-conditioned.
27	HappyCat	Multimodal/nonconvex BBOB-style function.
28	HGBat	Unimodal/nonconvex BBOB-style function.
29	HCF	Unimodal composition-style primitive.
30	SchafferF6	Multimodal.
31	Step	Unimodal stepped function.
32	Quartic	Unimodal quartic function.
33	Exponential	Unimodal exponential landscape.
34	StyblinskiTang	Multimodal.

Coordinate transforms

Let \(x\) be the parent point, \(a\) be the assigned component optimum stored in assigned_xopt, and \(t\) be the internally resolved child optimum. The transform maps parent coordinates into child coordinates.

none

Full-dimensional shift:

\[T(x) = t + (x-a).\]

rotation

Full-dimensional shifted rotation with orthogonal matrix \(R\):

\[T(x) = t + R(x-a).\]

affine

Full-dimensional shifted affine transform with matrix \(A\):

\[T(x) = t + A(x-a).\]

block-rotation

Subspace transform. If selected_indices defines projection \(P\), then \(x_{\mathrm{sub}} = Px\) and

\[T(x) = t + R(x_{\mathrm{sub}} - a).\]

The child function sees only output_dimension variables.

Value transforms

Let \(u \ge 0\) be the shifted component value before the value transform.

none

\[\phi(u) = u.\]

power

Parameters are [alpha, p]:

\[\phi(u) = \alpha u^p.\]

oscillatory

Parameters are [epsilon, alpha]:

\[\phi(u) = u\left(1 + \epsilon\sin(\alpha u)\right).\]

cosine-zero

Parameter is [alpha]:

\[\phi(u) = 1 - \cos(\alpha u).\]

Trigonometric value transforms reduce the phase modulo \(2\pi\) internally for more stable cross-platform numerical behavior.

Composition modes

Let \(z_i\) be transformed component values.

none

Single-component identity:

\[\psi(z_1) = z_1.\]

cpm-wsum

Common-point weighted sum:

\[\psi(z) = \sum_i w_i z_i.\]

cpm-power-mean

Parameter is [p]:

\[\psi(z) = \left(\sum_i w_i z_i^p\right)^{1/p}.\]

cpm-level-well

Parameters are [epsilon, alpha]. Let \(s = \sum_i w_i z_i\):

\[\psi(z) = s\left(1 + \epsilon\sin(\alpha s)\right).\]

dpm-softmax

Parameter is [sharpness]. Let \(c_i\) be full-dimensional DPM centers, \(b_i\) be DPM biases, and \(\gamma\) be sharpness:

\[q_i(x) = \exp(-\gamma\|x-c_i\|^2 - M), \qquad M = \max_j -\gamma\|x-c_j\|^2.\]

Non-global centers are masked near the global center:

\[m_0(x)=1,\qquad m_i(x)=1-\exp(-\|x-c_0\|^2),\quad i>0.\]

Then

\[\psi(x,z) = \frac{\sum_i q_i(x)m_i(x)(z_i+b_i)} {\sum_i q_i(x)m_i(x)}.\]

dpm-bgsoftmax

Parameters are [sharpness, background_strength, background_sharpness]. It adds a smooth background term

\[\beta(x) = \rho\left(1-\exp(-\eta \min_i\|x-c_i\|)\right)\]

and computes

\[\psi(x,z) = \frac{\sum_i (q_i(x)m_i(x)+\beta(x))(z_i+b_i)} {\sum_i (q_i(x)m_i(x)+\beta(x))}.\]

DPM center 0 is the assigned global optimum. Other centers are deceptive locations. DPM biases are composition parameters, not component value transforms.

Name aliases

Examples of accepted aliases:

Canonical name	Common aliases
none	identity
rotation	rot
affine	aff
block-rotation	blockrotation, blockrot, brot
cpm-wsum	cpmsum, weighted_sum
cpm-power-mean	cpmpmean, power_mean
cpm-level-well	cpmlwell, level_well
dpm-softmax	dpmsoftmax, dpm
dpm-bgsoftmax	dpmbgsoftmax, DPM BG Softmax