[Paper] Pascal-Weighted Genetic Algorithms: A Binomially-Structured Recombination Framework

Source: arXiv - 2512.01249v1

Overview

Otman A. Basir’s paper presents Pascal‑Weighted Genetic Algorithms (PWR) – a new family of multi‑parent crossover operators that blend several parents using binomial (Pascal‑triangle) weights. By shaping the inheritance probabilities, PWR reduces disruptive jumps while preserving useful building blocks, leading to faster, more reliable convergence across a range of optimisation problems.

Key Contributions

• Binomial‑structured recombination: Introduces a mathematically‑grounded way to assign convex‑combination weights derived from normalized Pascal coefficients.
• Variance‑transfer analysis: Derives closed‑form expressions showing how PWR dampens offspring variance compared with classic two‑parent crossover.
• Schema survival theory: Extends classic schema theorem to multi‑parent, weighted recombination, proving higher preservation of useful schemata.
• Representation‑agnostic extensions: Provides concrete formulations for real‑valued vectors, binary/logit encodings, and permutation chromosomes (e.g., TSP tours).
• Empirical validation on four diverse benchmarks: PID controller tuning, FIR filter design, wireless power‑modulation, and the Traveling Salesman Problem.
• Performance gains: Demonstrates 9‑22 % improvement in final objective quality and smoother convergence curves versus standard operators (single‑point, uniform, PMX, etc.).

Methodology

1. Weight Generation – For a chosen number of parents k, the algorithm computes normalized Pascal coefficients
   [ \hat{w}_i = \frac{\binom{k-1}{i-1}}{2^{k-1}} . ]
   These weights form a symmetric, bell‑shaped distribution that peaks at the centre of the parent list.

2. Offspring Construction –

   • Real‑valued:
     [ x^{\text{off}} = \sum_{i=1}^{k} \hat{w}_i , x^{(i)} . ]
   • Binary/Logit: Each gene is sampled from a Bernoulli distribution whose success probability is the weighted sum of parent bits.
   • Permutation: A weighted voting scheme selects the position of each element, followed by a repair step to enforce a valid tour.

3. Integration into GA Loop – PWR replaces the usual two‑parent crossover step; selection, mutation, and elitism remain unchanged, making the operator a drop‑in module for any existing GA framework.

4. Theoretical Analysis – Using linearity of expectation and properties of binomial coefficients, the author derives:

   • Expected offspring mean equals the mean of the parent pool (unbiased).
   • Offspring variance is reduced by a factor of (1/2^{k-1}) relative to uniform weighting, explaining the smoother search dynamics.

5. Experimental Setup – Each benchmark runs 30 independent GA trials (population = 200, generations = 500) comparing PWR (k = 3, 5) against standard crossover baselines. Performance is measured by domain‑specific metrics (ITAE for PID, magnitude‑error for FIR, SINR for wireless, tour length for TSP).

Results & Findings

Across all tests, PWR’s multi‑parent averaging yields lower offspring variance, which translates into steadier progress and a higher chance of preserving promising schemata. The gains are most pronounced when the search space is rugged or when the fitness landscape contains many local optima (e.g., TSP).

Practical Implications

• Plug‑and‑play improvement: Developers can swap their existing crossover routine for a PWR module with minimal code changes; the operator works with any GA library that supports custom recombination.
• Robustness for noisy or real‑time optimisation: The variance‑reduction property makes PWR attractive for control‑system tuning (PID) or online wireless‑resource allocation where fitness evaluations are noisy.
• Scalable to high‑dimensional problems: Because the weighted sum is linear, the computational overhead grows only with the number of parents, not with chromosome length, keeping runtime comparable to classic crossover.
• Better exploitation of parallelism: Multi‑parent selection naturally fits GPU/CPU batch processing—select k parents per offspring in a single kernel launch.
• Potential for hybrid meta‑heuristics: PWR can be combined with differential evolution, particle swarm, or memetic local search to provide a smoother global search backbone before fine‑grained exploitation.

Limitations & Future Work

• Parent count sensitivity: Diminishing returns beyond k = 5; too many parents can oversmooth the search, eroding diversity.
• Permutation repair cost: For large TSP instances, the repair step adds overhead; more efficient permutation‑specific weighting schemes could be explored.
• Theoretical bounds limited to simple fitness models: The variance analysis assumes additive fitness components; extending the theory to highly epistatic or constrained problems remains open.
• Future directions suggested include adaptive selection of k based on population diversity metrics, integration with self‑adaptive mutation rates, and testing on deep‑learning hyper‑parameter optimisation or neural architecture search.

Authors

• Otman A. Basir

Paper Information

• arXiv ID: 2512.01249v1
• Categories: cs.NE, cs.AI, eess.SY
• Published: December 1, 2025
• PDF: Download PDF