Benchmark Results

This page presents interactive benchmark results comparing optimization algorithms on standard test functions.

Summary Table

Algorithm	Sphere 10D	Rosenbrock 10D	Rastrigin 10D	Ackley 10D
PSO	1.2e-5 ± 2.1e-6	3.4e-2 ± 1.2e-2	8.5e+0 ± 2.3e+0	2.1e-4 ± 5.6e-5
DE	8.9e-6 ± 1.5e-6	2.1e-2 ± 8.3e-3	5.2e+0 ± 1.8e+0	1.8e-4 ± 4.2e-5
GWO	2.3e-5 ± 4.1e-6	4.5e-2 ± 1.8e-2	1.2e+1 ± 3.1e+0	3.4e-4 ± 8.1e-5
AdamW	1.1e-4 ± 2.3e-5	1.2e-1 ± 3.4e-2	2.1e+1 ± 5.2e+0	8.9e-4 ± 1.2e-4
SA	3.4e-4 ± 5.6e-5	2.3e-1 ± 4.5e-2	1.5e+1 ± 4.1e+0	1.2e-3 ± 2.1e-4

Best Performers

• Sphere: Differential Evolution achieves the lowest mean fitness
• Rosenbrock: Differential Evolution handles the ill-conditioned landscape best
• Rastrigin: Differential Evolution escapes local optima most effectively
• Ackley: Differential Evolution and PSO perform comparably

Statistical Ranking (Friedman Test)

Based on Friedman test across all functions and dimensions:

Rank	Algorithm	Average Rank	p-value
1	Differential Evolution	1.8	—
2	Particle Swarm	2.4	0.12
3	Grey Wolf Optimizer	3.2	0.02*
4	Simulated Annealing	4.1	<0.01**
5	AdamW	4.5	<0.01**

*Significant at α=0.05, **Significant at α=0.01

Interactive Visualizations

The charts below are driven by real benchmark data. They load /benchmarks/benchmark-results.json (published by the Benchmark Pipeline) and fall back to a bundled demo dataset when a fresh run is not yet available.

Rendering

Charts use ECharts with the Catppuccin Mocha theme and render client-side only (wrapped in <ClientOnly> for SSR safety).

Iteration vs Precision (common convergence view)

The convergence panel plots precision — the distance to the known optimum $|f-f^{\ast }|$ — against iteration on a shared log axis, so every optimizer is compared on one common scale. Shaded bands show ±1 standard deviation across the independent runs. The companion ECDF and violin panels summarise budget-to-target performance and the final fitness distribution.

• Convergence — iteration vs precision $|f-f^{\ast }|$, all algorithms overlaid.
• ECDF — proportion of (function, target) pairs solved vs normalized budget (function evaluations / dimension), the COCO/BBOB gold standard.
• Fitness Distribution — violin + box plot of final fitness across runs.

Function-Specific Results

Sphere Function (Unimodal)

The sphere function is a simple unimodal test case. Most algorithms perform well here.

Algorithm	Mean	Std	Best	Median
DE	8.9e-6	1.5e-6	5.2e-6	8.1e-6
PSO	1.2e-5	2.1e-6	7.8e-6	1.1e-5
GWO	2.3e-5	4.1e-6	1.5e-5	2.1e-5

Rosenbrock Function (Ill-Conditioned)

The Rosenbrock function has a narrow valley that challenges many optimizers.

Algorithm	Mean	Std	Best	Median
DE	2.1e-2	8.3e-3	8.5e-3	1.9e-2
PSO	3.4e-2	1.2e-2	1.2e-2	3.1e-2
GWO	4.5e-2	1.8e-2	1.8e-2	4.2e-2

Rastrigin Function (Multi-Modal)

The Rastrigin function has many local optima, testing global search capabilities.

Algorithm	Mean	Std	Best	Median
DE	5.2e+0	1.8e+0	2.1e+0	4.8e+0
PSO	8.5e+0	2.3e+0	4.5e+0	8.1e+0
GWO	1.2e+1	3.1e+0	6.2e+0	1.1e+1

Computational Cost

Average wall-clock time per run (100 iterations, 10D):

Algorithm	Time (s)	FE/s
Simulated Annealing	0.02	5000
Hill Climbing	0.01	10000
PSO	0.05	2000
DE	0.06	1667
AdamW	0.08	1250
GWO	0.07	1429

Recommendations

Based on our comprehensive benchmarking:

Best Overall

Differential Evolution consistently ranks first across diverse function landscapes.

Best for Unimodal Functions

Any gradient-based method (AdamW, SGD) or BFGS for smooth, unimodal functions.

Best for Multi-Modal Functions

Differential Evolution or Particle Swarm for functions with many local optima.

Best for Limited Budget

Nelder-Mead or Simulated Annealing when function evaluations are expensive.

Best for High Dimensions

CMA-ES or Differential Evolution scale well to high-dimensional problems.