Ackley Function (20D)

[MAL-Seeker Execution Record] Global Optimization of the 20D Ackley Function

$$f(\mathbf{x}) = -20 \exp \left( -0.2 \sqrt{\frac{1}{d} \sum_{i=1}^{d} x_i^2} \right) - \exp \left( \frac{1}{d} \sum_{i=1}^{d} \cos(2\pi x_i) \right) + 20 + e$$

d = 20 (20-Variable Dimension)
Global Minimum: $f(0, \dots, 0) = 0$

1. Problem Definition

Using the custom solver MAL-Seeker (V4.0.0), we performed a global optimization search on the "Ackley Function," a representative benchmark for global optimization, in a 20-dimensional space. The search range for each dimension was $-32.768 \le x_i \le 32.768$.

This function is characterized by an overall gentle gradient toward the center (origin), while its surface is covered with countless sharp local minima. It resembles a "bed of nails" terrain, where it is extremely difficult to reach the true center once an algorithm gets trapped in a small nearby valley.

2. Search Approach: Trap Avoidance via Five Senses + Sixth Sense

In the vast 20-dimensional search space, MAL-Seeker V4.0.0 captured the center point through the following processes:

・Memetic Crossover (Hearing): Blending coordinate information across multiple generations to condense global solution candidates into "super-agents."
・Relative Fuzzy Logic (Sixth Sense): Mass-purging local minima (the bed of nails) using dynamic thresholds based on the swarm's average score.
・AD-based Newton Method (Touch): Calculating an accurate Jacobian with the built-in Automatic Differentiation engine to achieve a zero-error landing at the origin.

3. Execution Results

Final log in the 20-dimensional space (MacBook environment, 1.2GHz Dual-Core):

=========================================

  🏆 BEST OPTIMIZATION RESULT 🏆
  Score (Objective) : -0.00000002
  Coordinates       : (-0.0000000, 0.0000000, -0.0000000, -0.0000000, -0.0000000, 0.0000000, -0.0000000, -0.0000000, 0.0000000, 0.0000000, -0.0000000, 0.0000000, -0.0000000, -0.0000000, -0.0000000, -0.0000000, 0.0000000, 0.0000000, -0.0000000, -0.0000000)
=========================================
=> CSV Export: Successfully saved to 'ackley_20d.txt_result.csv'

Execution Time: 313.761250 seconds

Dr.WataWata Insight:

While the Ackley function is relatively easy in 2D, the density of local minima becomes overwhelming in a 20D space, forcing most solvers into stagnation.
The key to success with V4.0.0 was the newly implemented "Sixth Sense." By using the swarm's average score as a yardstick, it ruthlessly dismissed the surrounding traps as "valueless fakes," maintaining a clear route to the center. Achieving a score near the limits of floating-point precision in about 5 minutes proves the robustness of this Relative Fuzzy Logic.

4. Analysis Environment

Analysis Engine: MAL-Seeker (V4.0.0 / C / 5SS_NEWTON)
PC: MacBook
OS: macOS Monterey 12.7.6
CPU: 1.2GHz Dual-Core Intel Core m5
Memory: 8GB

1. Problem Definition

2. Search Approach: Trap Avoidance via Five Senses + Sixth Sense

3. Execution Results

4. Analysis Environment

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