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

• URL: http://arxiv.org/abs/2607.29228v1
• Date: Fri, 31 Jul 2026 09:58:58 GMT
• Title: Linear Proposal Operators and Stochastic Search Geometry in SOMA and Differential Evolution
• Abstract summary: This paper introduces an operator--selection factorization that separates objective-independent variation from boundary repair and fitness-dependent selection.<n>It uses it to study the proposal geometry of the Self-Organizing Algorithm (SOMA) and Differential Evolution (DE)
• Score: 0.06875312133832077
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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