SOMA与差分进化中的线性提议算子与随机搜索几何

中文摘要 AI 辅助

群体与进化算法通常被视为完整的程序系统，其中非线性选择、替换与自适应操作掩盖了候选解生成过程中的简单结构。本文引入一种算子-选择分解方法，将与目标无关的变异、边界修复及依赖适应度的选择操作分离开来，并利用该方法研究自组织迁移算法（Self-Organizing Migrating Algorithm, SOMA）与差分进化（Differential Evolution, DE）的提议几何。研究表明，标准SOMA的提议在搜索空间中为仿射变换，且在增广的迁移个体-领导者状态下呈严格线性；在领导者相对坐标中，该算子可直接解释插值、投影、超调及坐标掩码现象。针对伯努利扰动掩码，本文推导了提议均值、协方差、期望平方步长、期望与领导者的平方距离、活跃维度及坐标覆盖度的闭式表达式；针对标准DE/rand/1/bin，推导了差分变异的有限种群矩，并表征了强制坐标二项式交叉诱导的额外协方差与坐标依赖性。精确枚举与蒙特卡洛实验验证了上述解析关系，并量化了掩码条件、边界修复及基于适应度的选择的影响。该分析进一步推动了几何控制与旋转感知的SOMA变体，以及iSOMA的自适应种群缩减扩展。在完整无噪声BBOB基准上的实验表明，这些受算子指导的变体显著优于标准SOMA，且在多个维度-预算 regime 下与成熟DE方法具有竞争力。结果表明，提议层面的算子分析可同时支持基于种群优化器的解释与设计。

英文摘要

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.

发表机构

• VSB - Technical University of Ostrava（俄斯特拉发技术大学（VSB））
• IT4Innovations National Supercomputing Center（IT4Innovations国家超级计算中心）
• Marine Research Institute, Klaipeda University（克莱佩达大学海洋研究所）

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