Geometric Thermodynamic Cockroach Optimization Algorithm Based on Dynamic Riemannian Metric and Non-Equilibrium Free Energy

Swarm optimization algorithms have wide applications in complex non-convex optimization problems, but traditional algorithms are usually based on the Euclidean space assumption, lacking adaptive characterization of the geometric structure of the search space, and also lacking a unified physical interpretation at the dynamic level. This paper proposes a cockroach optimization algorithm based on dynamic Riemannian metric, pheromone-energy coupling field, and non-equilibrium free energy driving mechanism. The algorithm constructs a dynamic metric matrix induced by the population covariance, enabling the search process to take place in a non-Euclidean manifold; it modulates the development intensity through the coupling of the pheromone field and individual energy variables; it models the algorithm as a non-equilibrium dissipative system by introducing population structural entropy and free energy function; and it triggers a population reconstruction mechanism under entropy collapse conditions to avoid premature convergence. This paper systematically derives the algorithm from the perspectives of geometric optimization, information theory, and statistical physics, and provides a discussion of the dynamic system form and stability. Theoretical analysis shows that the algorithm has stronger local convergence stability while maintaining global exploration capability.