Case Study: Multimodal Benchmarks¶

Beyond systems of equations, PySNE can locate many optima of a single objective function — the multimodal optimisation task. These benchmarks live in pysne.problems.benchmarks_multimodal and are the standard functions used to test niching and multi-solution optimisers (Rastrigin, Shubert, Six-Hump Camel Back, and more).

How this differs from the SNE benchmarks

In the SNE suite the goal is to solve \(F(\mathbf{x}) = 0\), and problems carry an expected_roots target. Here the goal is to find all optima of \(g(\mathbf{x})\), so each problem instead declares an optima_type:

"max" — search for maxima only
"min" — search for minima only
"both" — search for maxima and minima (the default)

The solver finds maxima directly; to find minima, the problem is wrapped in MinimizedProblem before solving. Success is measured by the number of optima found, not a fixed root count.

Running a multimodal problem¶

from pysne.problems.benchmarks_multimodal import get_multimodal_problems
from pysne.solver import solve_system

# Registry accepts integer IDs and named keys
problems = get_multimodal_problems()

# Six-Hump Camel Back (ID 2)
prob = problems[2]()
domain, params = prob.get_info()

result = solve_system(prob, params, verbose=True)
optima = result["roots"]

print(f"{prob.name}")
print(f"optima_type : {prob.optima_type}")
print(f"Optima found: {len(optima)}")

Finding both maxima and minima for a "both" problem:

from pysne.problems.benchmarks_multimodal import get_multimodal_problems
from pysne.problems.base import MinimizedProblem
from pysne.solver import solve_system

prob = get_multimodal_problems()[1]()   # Two-N-Minima, optima_type = "both"
_, params = prob.get_info()

maxima = solve_system(prob, params, verbose=False)["roots"]
minima = solve_system(MinimizedProblem(prob), params, verbose=False)["roots"]

print(f"Maxima: {len(maxima)} | Minima: {len(minima)}")

Benchmark roster¶

The registry exposes both numeric IDs and convenient string keys.

Key	Function	Vars	Search space	optima_type
`1` / `two_n_minima`	Two-N-Minima	2	\([-4, 4]^2\)	both
`2`	Six-Hump Camel Back	2	\([-1.9, 1.9] \times [-1.1, 1.1]\)	both
`3` / `rastrigin`	Rastrigin	2	\([-1, 1]^2\)	both
`4`	Rastrigin	3	\([-1, 1]^3\)	both
`5`	Vincent	2	\([0.25, 10]^2\)	max
`6`	Shubert	2	\([-10, 10]^2\)	max
`7`	Shubert	3	\([-10, 10]^3\)	max
`schwefel`	Schwefel 2.22	3	\([-4, 6]^3\)	both
`griewank`	Griewank	2	\([-600, 600]^2\)	both
`iwm`	IWM project (applied)	25	task-specific bounds	both

Two-N-Minima¶

A separable quartic whose one-dimensional profile has two minima per axis, giving a small, well-understood set of optima — a good first sanity check.

\[ g(\mathbf{x}) = \sum_{i} \frac{1}{2}\left(x_i^4 - 16 x_i^2 + 5 x_i\right) \]

Search space: \(x_1, x_2 \in [-4, 4]\)
optima_type: both

Six-Hump Camel Back¶

A canonical low-dimensional multimodal test function with six local optima, two of which are global minima.

\[ g(\mathbf{x}) = \left(4 - 2.1 x_1^2 + \frac{x_1^4}{3}\right) x_1^2 + x_1 x_2 + \left(-4 + 4 x_2^2\right) x_2^2 \]

Search space: \(x_1 \in [-1.9, 1.9]\), \(x_2 \in [-1.1, 1.1]\)
optima_type: both

Rastrigin (2D / 3D)¶

A highly multimodal function with a regular lattice of local minima — the standard stress test for niching. Available in 2D (3) and 3D (4) variants over a compact box.

\[ g(\mathbf{x}) = \sum_{i} \left(x_i^2 - 10 \cos(2\pi x_i) + 10\right) \]

Search space: \(x_i \in [-1, 1]\)
optima_type: both

Vincent¶

An oscillatory function on a log-scaled domain, with maxima whose spacing shrinks toward the lower bound.

\[ g(\mathbf{x}) = \frac{1}{2}\left(\sin(10 \ln x_1) + \sin(10 \ln x_2)\right) \]

Search space: \(x_1, x_2 \in [0.25, 10]\)
optima_type: max

Shubert (2D / 3D)¶

A separable product of cosine sums that produces a large number of global maxima — one of the hardest counting tasks in the suite. Provided in 2D (6) and 3D (7).

\[ g(\mathbf{x}) = -\prod_{j} \sum_{i=1}^{5} i \cos\left((i + 1) x_j + i\right) \]

Search space: \(x_j \in [-10, 10]\)
optima_type: max

Schwefel 2.22¶

A convex-but-nonsmooth function combining an absolute-value sum with a product term; its single minimum at the origin makes it a useful precision check in higher dimensions.

\[ g(\mathbf{x}) = \sum_{i} |x_i| + \prod_{i} |x_i| \]

Search space: \(x_i \in [-4, 6]\) (3D)
optima_type: both

Griewank¶

A function with many regularly spaced local minima over a very wide domain, testing robustness to scale.

\[ g(\mathbf{x}) = \sum_{i} \frac{x_i^2}{4000} - \prod_{i} \cos\left(\frac{x_i}{\sqrt{i}}\right) + 1 \]

Search space: \(x_1, x_2 \in [-600, 600]\)
optima_type: both

IWM (applied problem)¶

An applied 25-dimensional case in which each variable is a task duration bounded by project-specific limits, showing how the same multimodal machinery extends to real planning problems. Retrieve it with the "iwm" key.

Method¶

Multimodal problems reuse PySNE's Clustering → SPO → Selection pipeline, with the selection stage adapted for optima rather than roots: candidates are filtered against a global threshold derived from the best objective value found (\(F^*\)) and the gamma parameter, then merged by proximity (\(\delta\)). See Algorithms for details.