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Pascal-Weighted Genetic Algorithms: A Binomially-Structured Recombination Framework

Otman A. Basir

TL;DR

This work introduces Pascal-Weighted Recombination (PWR), a principled multi-parent crossover that uses binomial (Pascal) weights to form offspring as convex blends of multiple parents. By connecting to Bernstein polynomials, PWR provides a smooth, variance-controlled inheritance mechanism that preserves schemas and reduces disruptive jumps across real-valued, binary/logit, and permutation domains. Theoretical analysis yields a closed-form variance contraction that strengthens with the number of parents, and practical algorithms are provided for integrating PWR into standard GA pipelines. Empirical evaluations across PID tuning, FIR filter design, wireless SINR optimization, and the TSP show that PWR achieves smoother convergence, lower variance, and superior solution quality compared with traditional two-parent and other multi-parent operators. The results support PWR as a simple, versatile building block for variance-aware evolutionary optimization with broad applicability in engineering and combinatorial problems.

Abstract

This paper introduces a new family of multi-parent recombination operators for Genetic Algorithms (GAs), based on normalized Pascal (binomial) coefficients. Unlike classical two-parent crossover operators, Pascal-Weighted Recombination (PWR) forms offsprings as structured convex combination of multiple parents, using binomially shaped weights that emphasize central inheritance while suppressing disruptive variance. We develop a mathematical framework for PWR, derive variance-transfer properties, and analyze its effect on schema survival. The operator is extended to real-valued, binary/logit, and permutation representations. We evaluate the proposed method on four representative benchmarks: (i) PID controller tuning evaluated using the ITAE metric, (ii) FIR low-pass filter design under magnitude-response constraints, (iii) wireless power-modulation optimization under SINR coupling, and (iv) the Traveling Salesman Problem (TSP). We demonstrate how, across these benchmarks, PWR consistently yields smoother convergence, reduced variance, and achieves 9-22% performance gains over standard recombination operators. The approach is simple, algorithm-agnostic, and readily integrable into diverse GA architectures.

Pascal-Weighted Genetic Algorithms: A Binomially-Structured Recombination Framework

TL;DR

This work introduces Pascal-Weighted Recombination (PWR), a principled multi-parent crossover that uses binomial (Pascal) weights to form offspring as convex blends of multiple parents. By connecting to Bernstein polynomials, PWR provides a smooth, variance-controlled inheritance mechanism that preserves schemas and reduces disruptive jumps across real-valued, binary/logit, and permutation domains. Theoretical analysis yields a closed-form variance contraction that strengthens with the number of parents, and practical algorithms are provided for integrating PWR into standard GA pipelines. Empirical evaluations across PID tuning, FIR filter design, wireless SINR optimization, and the TSP show that PWR achieves smoother convergence, lower variance, and superior solution quality compared with traditional two-parent and other multi-parent operators. The results support PWR as a simple, versatile building block for variance-aware evolutionary optimization with broad applicability in engineering and combinatorial problems.

Abstract

This paper introduces a new family of multi-parent recombination operators for Genetic Algorithms (GAs), based on normalized Pascal (binomial) coefficients. Unlike classical two-parent crossover operators, Pascal-Weighted Recombination (PWR) forms offsprings as structured convex combination of multiple parents, using binomially shaped weights that emphasize central inheritance while suppressing disruptive variance. We develop a mathematical framework for PWR, derive variance-transfer properties, and analyze its effect on schema survival. The operator is extended to real-valued, binary/logit, and permutation representations. We evaluate the proposed method on four representative benchmarks: (i) PID controller tuning evaluated using the ITAE metric, (ii) FIR low-pass filter design under magnitude-response constraints, (iii) wireless power-modulation optimization under SINR coupling, and (iv) the Traveling Salesman Problem (TSP). We demonstrate how, across these benchmarks, PWR consistently yields smoother convergence, reduced variance, and achieves 9-22% performance gains over standard recombination operators. The approach is simple, algorithm-agnostic, and readily integrable into diverse GA architectures.

Paper Structure

This paper contains 34 sections, 34 equations, 8 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Research Gap and Contributions
  3. Related Work
  4. Two-Parent Crossover
  5. Multi-Parent Recombination
  6. Variance-Controlled Recombination
  7. Gap in the Literature
  8. Methodology
  9. Variance and Diversity Properties
  10. Algorithmic Integration
  11. Extension to Real-Coded and Combinatorial Domains
  12. Pascal-Weighted Genetic Algorithms: Theory
  13. Connection to Bernstein Polynomials
  14. Schema Survival Analysis
  15. Representation-Specific Variants
  16. ...and 19 more sections

Figures (8)

  • Figure 1: (a) Pascal-binomial weighting patterns and (b) corresponding variance reduction behavior in PWR.
  • Figure 2: Geometric foundation of Pascal-Weighted Recombination (PWR).
  • Figure 3: PID controller tuning: (a) evolutionary convergence under baseline GA vs. Pascal-weighted GA (PWR-3), and (b) corresponding closed-loop step responses for best individuals.
  • Figure 4: Pascal-weight structure (a) and resulting variance behavior (b) under Pascal-Weighted Recombination.
  • Figure 5: Convergence behavior for the wireless resource allocation problem: baseline GA vs. Pascal-weighted GA (PWR-3). Higher utility is better.
  • ...and 3 more figures