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

Integer Natural Evolution Strategies

While contemporary Evolution Strategies handle integer optimization problems effectively, their adaptation mechanism is grounded in $\ell_2$-based Gaussian models, which are not native to the integer lattice. In contrast, the $\ell_1$-norm provides the natural measure of displacement on $\mathbb{Z}^n$, with the double geometric distribution as its canonical mutation operator. In this work, we derive a fully $\ell_1$-native step-size adaptation mechanism from first principles and propose an Integer Natural Evolution Strategy. We show that the DG distribution belongs to the exponential family, and that its sufficient statistic $|z|$ yields a natural-gradient signal for dispersion adaptation. By accumulating this signal via an evolution path, we obtain a fading-memory online estimator of the natural gradient, following Ollivier (2018). This establishes that DG-based step-size adaptation arises directly from the statistical structure of the mutation distribution, rather than as a discrete analog of continuous ES mechanisms. Empirical results on integer quadratic benchmarks show that \textsc{INES} learns meaningful coordinate-wise step-sizes and is competitive with integer-handling CMA-ES baselines. Its advantages are most visible in high-dimensional Ellipsoidal problems and in robust convergence at larger dimensions.

Long-time Stability and Convergence of Particle Swarm Optimization

Particle Swarm Optimization (PSO) is a global optimization algorithm defined by an interacting set of particles evolving over the search space. Heuristically motivated, its theoretical analysis remains limited due to the second-order, stochastic, and highly nonlinear nature of the dynamics. In this paper, we connect classical PSO stability analysis under the stagnation assumption with more recent mean-field methods, providing new quantitative estimates for the time-discrete algorithm. We study in particular a regularized PSO model without memory, with non-degenerate noise by adding a noise floor to the original model. Studying such a surrogate model allows us to identify quantitative conditions under which the dynamics is stable and converges toward a small neighborhood of a global minimizer. We do so by first studying the Schur stability of the linearized dynamics, then analyzing the convergence properties of a nonlinear mean-field system via a Laplace principle, and finally establishing a quantitative error bound for the mean-field approximation of order $N^{-1/2}$.

JANUS: Online Jacobian-Aligned Infill for Black-Box Optimization

JANUS is a plug-and-play infill module that extracts a local Jacobian from the recent evaluation trace, and gives the best mean cost on 1135-dimensional UAV path planning and improves SMS-EMOA/AGE-MOEA2 hosts on 12/38 multi-objective tasks with zero significant regressions.

Differential evolution with fitness-based mutation strategy and diversity enhancement mechanism for numerical optimization

Non-Gaussianity of the Stagnation Law in Particle Swarm Optimization

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 \[ -10,\qquad 12(1-w^2)-c(7-5w)>0, \] 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.

An Improved Attraction-Repulsion OptimizationAlgorithm for Global Optimization and EngineeringDesign Problems

Attraction-Repulsion Optimization Algorithm (AROA) is a recently proposed meta-heuristic algorithm known for its simplicity, ease of implementation, and robustness. However, AROA may converge to local optima when applied to complex optimization problems. To address this limitation, we propose an enhanced version called the Differential Cauchy Tangent Attraction-Repulsion Optimization Algorithm (DCTAROA). First, we propose a mutation operator based on a tangent flight mutation strategy and a dimension decision mechanism using the inverse cumulative distribution function of the Cauchy distribution. The tangent flight mutation enhances the local search capability and accelerates convergence, while the dimension decision strategy of the Cauchy distribution inverse cumulative function increases population diversity and improves exploration efficiency. Subsequently, we integrate Differential Evolution (DE) as a local search mechanism to strengthen the global optimization performance of AROA. To evaluate the proposed algorithm, we compare it with 15 state-of-the-art algorithms on 29 CEC2017 benchmark functions across various dimensions. Experimental results demonstrate that DCTAROA outperforms the compared algorithms in terms of solution accuracy, stability, convergence speed, and statistical significance based on the Wilcoxon rank-sum test. Furthermore, we apply DCTAROAto three practical engineering design problems. The results confirm that DCTAROA effectively explores the search space and yields competitive solutions, thereby validating its practical applicability.