粒子群优化中停滞定律的非高斯性
Non-Gaussianity of the Stagnation Law in Particle Swarm Optimization
AI总结：
研究一维粒子群优化停滞时位置分布，在特定均方稳定区域内，证明不存在高斯不变及极限位置边缘分布，解决相关开放问题，通过多阶矩方程比较及多项式分析证明，提供了相关数据和代码。
AI中文摘要：
我们研究在停滞期间具有两个固定不同吸引子且独立均匀加速度范围相等的一维粒子群优化。位置满足二阶随机仿射递推。对于惯性\(w\)和加速度范围\(c\)，我们证明在开放的均方稳定区域\(-1 < w < 1\)，\(c > 0\)，\(12(1 - w^2) - c(7 - 5w) > 0\)内，不存在不变位置边缘分布，因此也不存在极限位置边缘分布是高斯分布的情况。这解决了文献[粒子群问题]中的开放问题18。通过比较平稳矩方程和高斯矩恒等式到八阶来证明。使用埃尔米特多项式公式给出显式四阶和六阶兼容性条件，其共同解位于107次多项式分支上。精确的八阶方程排除了该分支上的每个点。最终证明通过模23和独立模1000003的算术验证。单独的原始矩实现产生关于\((w,c)\)的精确多项式\(Q_4\)、\(Q_6\)、\(Q_8\)，并在有理数上验证相同的障碍。四阶和六阶曲线有一个真正允许的交点，但八阶条件消除了它，说明了为什么低阶高斯诊断是不够的。所有代码、精确多项式、日志和绘图验证数据都作为在线资源提供。
英文摘要：
We study one-dimensional particle swarm optimization during stagnation, with two fixed distinct attractors and equal independent uniform acceleration ranges. The position then satisfies a second-order random affine recurrence. For inertia $w$ and acceleration range $c$, we prove that throughout the open mean-square stability region \[ -1<w<1,\qquad c>0,\qquad 12(1-w^2)-c(7-5w)>0, \] no invariant position marginal, and hence no limiting position marginal, can be Gaussian. This solves the open Problem 18 in \cite{ParticleSwarmProblems}. The proof compares the stationary moment equations with the Gaussian moment identities through order eight. A Hermite-polynomial formulation gives explicit fourth- and sixth-order compatibility conditions whose common solutions lie on a degree-$107$ polynomial branch. Exact eighth-order equations exclude every point on that branch. The final certificate is verified using arithmetic modulo $23$ and independently modulo $1{,}000{,}003$. A separate raw-moment implementation produces exact polynomials $Q_4,Q_6,Q_8$ in $(w,c)$ and verifies the same obstruction over the rational numbers. The fourth- and sixth-order curves have a genuine admissible intersection, but the eighth-order condition removes it, showing why low-order Gaussian diagnostics are insufficient. All code, exact polynomials, logs, and plot-validation data are supplied as online resources.
