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Schwefel Function (5D)

[MAL-Seeker Execution Record] Global Optimization of the 5D Schwefel Function

$$f(\mathbf{x}) = 418.9829d - \sum_{i=1}^{d} x_i \sin(\sqrt{|x_i|})$$

d = 5 (5-Variable Dimension)
Global Minimum: $f(420.9687, \dots, 420.9687) = 0$

1. Problem Definition

Using the custom numerical analysis solver MAL-Seeker (Ver 3.1.7), we performed global optimization on the standard benchmark "Schwefel function" in a 5-dimensional space. The search range for each dimension was $-500 \le x_i \le 500$.

This function is characterized by a structure known as "Deceptiveness." Since numerous deep local minima are located near the origin, many algorithms easily misidentify those points as the optimal solution. Furthermore, the true global minimum is located near the boundaries, making it notoriously difficult to capture.

2. Search Approach: Space Organization via Deflation

MAL-Seeker approached this problem through the following steps:

  • Multi-point Simultaneous Search: 100,000 initial seeds were placed based on the Latin Hypercube sampling method.
  • Deflation (Sense of Smell): Applied mathematical penalties to the surroundings of discovered local minima to prevent redundant searches.
  • Convergence via Automatic Differentiation: Utilized the built-in Automatic Differentiation (AD) engine to obtain accurate gradient information, improving convergence precision.

3. Execution Results

Execution results log in a MacBook environment (1.2GHz Dual-Core):

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

  🏆 BEST OPTIMIZATION RESULT 🏆
  Score (Objective) : 0.00006364
  Coordinates       : (420.9687463, 420.9687464, 420.9687464, 420.9687463, 420.9687464)
=========================================
=> CSV Export: Successfully saved to 'schwefel_5d.txt_result.csv'

Execution Time: 63.688999 seconds
Dr.WataWata Insight:

I believe that properly "excluding" the numerous local minima scattered in the central region through the deflation method led to the successful capture of the true optimal solution located at the periphery.
Reaching the global minimum in a 5D space within a practical timeframe without special tuning suggests that a search algorithm combining automatic differentiation and penalty functions holds significant validity for problems with such complex terrains.

4. Analysis Environment

  • Engine: MAL-Seeker (Antares Ver 3.1.7 / C / AD Engine)
  • PC: MacBook
  • OS: macOS Monterey 12.7.6
  • CPU: 1.2GHz Dual-Core Intel Core m5
  • Memory: 8GB