Conjugate schema and basis representation of crossover and mutation operators

Area of Science:

• Computational intelligence
• Optimization algorithms
• Mathematical foundations of computer science

Background:

• Genetic search algorithms and optimization routines often default to canonical bases for mutation and crossover operators.
• This default representation can limit the effectiveness and usefulness of these search algorithms.
• Identifying an optimal basis is crucial for maximizing algorithm performance.

Purpose of the Study:

• To investigate the impact of basis representation on genetic search algorithm utility.
• To introduce and define the conjugate schema as a novel mathematical construct.
• To demonstrate the performance improvements offered by conjugate schema in optimization tasks.

Main Methods:

• Introduction of the conjugate schema as a general mathematical construct.
• Demonstration of conjugate schema's ability to decompose functions into lower-dimensional components.
• Empirical evaluation using a test suite of 12 functions to assess performance improvements.

Main Results:

• The study shows that the choice of basis representation significantly influences the usefulness of genetic search algorithms.
• The conjugate schema effectively separates functions into summable, smaller dimensional functions.
• Application of conjugate schema resulted in improved performance for 10 out of 12 tested functions.

Conclusions:

• The canonical basis is not always optimal for genetic search algorithms.
• Conjugate schema provides a mathematically sound method for improving search algorithm performance.
• This approach offers a promising direction for enhancing the efficiency and effectiveness of genetic algorithms in optimization.