Hierarchical optimisation problems in volatile environments

• Applicant: Professor Mathias Staudigl, Ph.D.
• Subject Area: Mathematics ; Theoretical Computer Science
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 556222748

Hierarchical optimization problems arise in many applications in economics and engineering sciences. This class of non-convex optimization problems has traditionally been analyzed primarily in deterministic and stationary scenarios. To date, a systematic approach to solving hierarchical optimization problems in data-driven models remains an open question. The central challenge lies in developing efficient numerical methods for hierarchical optimization problems when data are learned sequentially, or when optimization problems explicitly depend on time. Solving such non-stationary hierarchical optimization problems provides us with new insights into the complexity of stochastic and simulation-based optimization problems. We also obtain new procedures for efficiently solving large-scale optimization problems with dynamic and probabilistic constraints. To be able to solve this important class of optimization problems, innovative new methods for solving complex hierarchical problems must be developed. This project is dedicated precisely to this question. In particular, we consider bilevel optimization problems in stochastic and volatile environments. The goal is to develop scalable numerical methods with explicit convergence rates for all levels. The focus is on efficiency and the ability to make decisions in real-time. The work program will develop solid mathematical theories and computational tools, whose performance will be demonstrated in several examples from machine learning and PDE-constrained optimization. In particular, we present in this work a new approach to bilevel optimization for solving optimization problems with probabilistic constraints. This class of problems plays a significant role in optimization problems where state constraints, modeled by dynamic processes (ODE or PDE), are incompletely specified or explicitly influenced by latent random processes.

• DFG Programme: Research Grants
• International Connection: Israel
• International Co-Applicant: Professorin Dr. Shimrit Shtern