Interacting particle systems for constrained optimization

Abstract

This thesis investigates two approaches to constrained optimization: particle swarm optimization (PSO) and a second-order Kalman-Langevin method. Both techniques are formulated in continuous time as stochastic differential equations (SDEs) and are implemented using numerical discretization schemes. Constraints are enforced through a reflection mechanism applied at the domain boundaries. The performance of the PSO algorithm is first evaluated on the Cross-in tray and Eggholder benchmark functions, where it demonstrates reliable convergence to global minima despite the highly non-convex landscapes. The Kalman-Langevin method is primarily assessed on the Rastrigin function, exhibiting robust performance in the presence of numerous local minima. To assess the applicability of these methods to practical problems, both algorithms are subsequently applied to the task of determining the optimal placement of a single gold atom on a gold surface, a problem characterized by a computationally expensive potential energy surface. The results indicate that both PSO and the Kalman-Langevin approach are effective in this setting, highlighting their generalizability beyond standard test functions. Furthermore, the parameter configurations identified during benchmark tuning are found to be transferable to the real-world application. These findings suggest that interacting particle systems governed by SDEs, such as PSO and Kalman-Langevin dynamics, constitute promising frameworks for addressing constrained optimization problems.