Particle methods for infinite dimensional optimization

• Applicants: Professor Dr. Massimo Fornasier; Professor Dr. Michael Herty; Professorin Dr. Claudia Totzeck
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 561130572

We aim at establishing a theoretical and numerical pathway for extending consensus-based optimization toward infinite dimensional problems. This will be tackled at three levels, namely, optimization problems with infinite number of minimizers, infinitely many parameters, or infinite-dimensional search domains. Consensus-based optimization is a derivative-free particle method, which has been developed for solving finite-dimensional global nonconvex optimization problems, allowing for rigorous theoretical guarantees. Building upon this established theory, we extend the concept of consensus-based optimization toward infinite dimensions leading to novel mathematical challenges, as for example, lacking compactness, a suitable definition of weighted center of mass in metric spaces, or the generalization of the dynamics to cope with parametric problems and manifolds of minimizers. The key tools and techniques we are going to employ come from kinetic theory, large deviation bounds, and efficient multilevel approximations for discretization and optimization.

• DFG Programme: Research Grants