Interacting particle system for nonconvex optimization

Collective Intelligence offers profound insights into how groups, whether they be cells, animals, or even machines, can work together to accomplish tasks more effectively than individuals alone. Originating in biology and now influencing fields as varied as management science, artificial intelligence, and robotics, this concept underscores the potential of collaborative efforts in solving complex challenges. On the other hand, the quest for finding global minimizers of nonconvex optimization problems arises in physics and chemistry, as well as in machine learning due to the widespread adoption of deep learning. Building the bridge between these two seemingly disparate realms, this project will utilize Collective Intelligence to leverage the interacting particle systems as a means to address the formidable challenge of finding global minimizers in nonconvex optimization problems. Graduate students will also be integrated within the research team as part of their professional training. This project will focus on a gradient-free optimization method inspired by a consensus-based interacting particle system to solve different types of nonconvex optimization problems. Effective communication and cooperation among particles within the system play pivotal roles in efficiently exploring the landscape and converging to the global minimizer. Aim 1 targets nonconvex optimization with equality constraints; and Aim 2 addresses nonconvex optimization on convex sets; while Aim 3 applies to Clustered Federated Learning. Additionally, convergence guarantees will be provided for nonconvex and nonsmooth objective functions. Theoretical analyses, alongside practical implementations, will provide valuable insights and tools for addressing different types of nonconvex optimization challenges. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.