Linear Proposal Operators and Stochastic Search Geometry in SOMA and Differential Evolution

2026-07-31

AI Summary

This work addresses the opacity of search dynamics in traditional swarm intelligence and evolutionary algorithms, where nonlinear selection and adaptive mechanisms obscure the intrinsic geometric structure underlying candidate solution generation. To resolve this, the authors propose an operator–selection decomposition framework that decouples fitness-independent variation operators from boundary handling and fitness-dependent selection, thereby systematically uncovering the proposal geometry of SOMA and Differential Evolution (DE). They reveal, for the first time, that SOMA exhibits a linear structure in the migrant–leader space, derive closed-form expressions for key statistical quantities, and leverage these insights to design a geometrically controllable, rotation-aware SOMA variant along with an adaptive population reduction strategy. On the noiseless BBOB benchmark suite, the resulting algorithm significantly outperforms the original SOMA and matches or exceeds state-of-the-art DE variants across multiple dimensions and evaluation budgets.

Abstract

Swarm and evolutionary algorithms are usually analyzed as complete procedural systems in which nonlinear selection, replacement, and adaptation obscure simpler structure within candidate generation. This paper introduces an operator--selection factorization that separates objective-independent variation from boundary repair and fitness-dependent selection, and uses it to study the proposal geometry of the Self-Organizing Migrating Algorithm (SOMA) and Differential Evolution (DE). The canonical SOMA proposal is shown to be affine in the search space and exactly linear in an augmented migrant--leader state. In leader-relative coordinates, the resulting operator provides a direct interpretation of interpolation, projection, overshooting, and coordinate masking. Under Bernoulli perturbation masks, we derive closed-form expressions for the proposal mean, covariance, expected squared step length, expected squared distance from the leader, active dimensionality, and coordinate coverage. For canonical DE/rand/1/bin, we derive the finite-population moments of differential mutation and characterize the additional covariance and coordinate dependence induced by forced-coordinate binomial crossover. Exact enumeration and Monte Carlo experiments verify the analytical identities and quantify the effects of mask conditioning, boundary repair, and fitness-based selection. The analysis further motivates geometry-controlled and rotation-aware SOMA variants, together with an adaptive population-reducing extension of iSOMA. Experiments on the complete noiseless BBOB benchmark show that these operator-guided variants substantially improve upon canonical SOMA and are competitive with established DE methods in several dimension--budget regimes. The results demonstrate how proposal-level operator analysis can support both the interpretation and design of population-based optimizers.

Problem

• proposal geometry
• swarm algorithms
• evolutionary algorithms
• operator analysis
• search space structure

Innovation

• operator–selection factorization
• proposal geometry
• linear proposal operator
• stochastic search analysis
• adaptive population reduction