## Ongoing Projects

Formal proof verification can both ensure proof correctness and provide new tools and insights to mathematicians. The goals of this project include creating resources for students and researchers, verifying relevant results, improving proof tactics, and exploring Machine Learning approaches.

This project will explore to what extent near-term quantum computing may potentially provide better approximations to combinatorial optimization problems, bringing together expertise from quantum computing and applied mathematics. It is a collaborative effort within the Einstein Research Unit on Quantum Devices, involving the Freie Universität Berlin (FU Berlin), the Weierstrass Institute for Applied Analysis and Stochastics (WIAS), and the Zuse Institute Berlin (ZIB).

Preserving forests is crucial for climate adaptation and mitigation. Accurate, up-to-date forest health data is essential. AI4Forest aims to develop advanced AI methods to monitor forests using satellite, radar, and LiDAR data. The project will create scalable techniques for detailed, high-resolution forest maps, updated weekly across Europe and globally.

We will apply a Frank-Wolfe-based approach for separability certification and entanglement detection of multipartite quantum states. The method will be further exploited to derive entanglement witnesses in the case, often encountered in experiments, of incomplete characterisation of the quantum state.

We will investigate mixed-integer optimization with convex objectives using error-adaptive convex solvers in branch-and-bound. Focusing on improving lower bounds and balancing computational costs, we aim to develop a faster branch-and-bound methodology by leveraging modern MILP techniques and error-adaptive methods. Key aspects include warm-starting and controlled inexactness in early termination.

This project aims to obtain new bounds in Extremal Combinatorics through an application of flag algebras. The goal is to both improve the underlying computational aspects for existing problems as well as to further develop the theory of flag algebras to extend it to new areas of application.

Motivated by nonsmooth problems in machine learning, we solve the problem of minimizing an abs-smooth function subject to closed convex constraints. New theory and algorithms are developed using linear minimization oracles to enforce constraints and abs-linearization methods to handle nonsmoothness.

Existing approaches for interpreting Neural Network classifiers that highlight features relevant for a decision are based solely on heuristics. We introduce a theory that allows us to bound the quality of the features without assumptions on the classifier model by relating classification to Interactive Proof Systems.

SynLab researches mathematical generalization of application-specific advances achieved in the Gas-, Rail– and MedLab of the research campus MODAL. The focus is on exact methods for solving a broad class of discrete-continuous optimization problems. This requires advanced techniques for structure recognition, consideration of nonlinear restrictions from practice, and the efficient implementation of mathematical algorithms on modern computer architectures. The results are bundled in a professional software package and complemented by a range of high-performance methods for specific applications with a high degree of innovation.

Computational devices and quantum mechanics revolutionized the 20th century. Quantum computation merges these fields to solve optimization problems faster than classical computers. This project aims to develop new quantum algorithms for general-purpose and specific applications, including optimization, mixed-integer programs, and uses in machine learning, logistics, big data, and physics. We will explore quantum dynamic programming, graph sparsification, and QAOA-type algorithms for small quantum computers.

In this project, we study domain decomposition approaches for optimal control in gas transport networks. Our goal is to couple space-time-domain decomposition with machine learning and mixed-integer programming. We will develop NeTI (Network Tearing and Interconnection), a data-driven and physics-informed algorithm combining mixed-integer nonlinear programming, surrogate model learning, and graph decomposition strategies.

## Completed Projects

Heuristics play a crucial role in exact solvers for Mixed Integer Programming (MIP). However, the question of how to manage multiple MIP heuristics in a solver has not received sufficient attention. This project addresses the strategic management of primal heuristics in MIP solvers, aiming to replace static, hard-coded rules with dynamic, self-improving procedures.

Airplane data quality is uneven due to varied sources and sensor limitations. This project aims to create data processing services for Component Spotting to help airlines optimize business processes and improve customer satisfaction. We will develop methods for modeling and evaluating data quality, creating a semantic and uniform description for processing models. Using AI algorithms, we will handle inaccuracies and uncertainties to form an analysis-friendly information model.

Worst-case complexity bounds are increasingly insufficient to explain the (often superior) real-world performance of optimization and learning algorithms. We consider data-dependent rates, approximation guarantees, and complexity bounds to provide guarantees much more in line with actual performance.

In this project, we study algorithms that promote sparsity. We develop PageRank optimization algorithms that scale with solution sparsity and investigate Riemannian optimization using manifold geometry. Additionally, we develop algorithms for efficient fair resource allocation based on established fairness axioms.

We develop theory and algorithms for 0-1 decision making in optimization problems constrained by partial differential equations. By exploring extended formulations, we achieve new stationarity concepts through sequential exact and approximative relaxation of adjoint-based primal-dual optimality conditions.

Extremal Combinatorics focuses on the maximum or minimum sizes of discrete structures with specific properties, posing significant challenges due to their complexity. Traditional computational approaches often fail due to exponential growth in search spaces, but recent AI advancements, especially in Reinforcement Learning, offer new potential. Applying these AI methods could provide insights into combinatorial problems while also enhancing the understanding of AI techniques in complex, sparse reward environments.

The performance of modern mixed-integer program solvers is highly dependent on a number of interdependent individual components. Using tools from machine learning, we intend to develop an integrated framework that is able to capture interactions of individual decisions made in these components with the ultimate goal to improve performance.

MiniMIP is an open source, machine learning oriented Mixed-Integer Programming (MIP) solver. We provide a range of interfaces for all aspects of solving MIPs (e.g. heuristics, cut generators, LP solvers), supplying users with a constant view of the internal state and allowing them to propose modifications that are integrated into the global state internally.

Quantum computing has the potential to occupy an important position in the field of computing in the long term. In the short to medium term, however, we can expect the emergence of so-called "Noisy Intermediate-Scale Quantum algorithms" and "Quantum Approximate Optimization Algorithms" in an initial phase. These algorithms can often solve a special subclass of problems approximately and very quickly using quantum computing and are significantly easier to implement technically. In this pilot project, a workflow will be tested, ranging from the design of hybrid quantum-classical algorithms to their simulation using quantum simulators on HPC hardware.

Deep learning is revolutionizing real-world applications and science, replacing or complementing classical model-based methods in solving mathematical problems. Despite successes, deep neural networks lack strong theoretical-mathematical foundations. This program aims to develop a comprehensive theoretical foundation of deep learning from three perspectives: statistical, application, and mathematical-methodological. The research is interdisciplinary, combining mathematics, statistics, and theoretical computer science to address complex questions.

Training artificial neural networks is a key optimization task in deep learning. To improve generalization, robustness, and explainability, we aim to compute globally optimal solutions. We will use integer programming methods, exploiting mixed-integer nonlinear programming and enhancing solving techniques like spatial branch-and-cut. Additionally, we'll leverage symmetry to reduce computational burden and ensure symmetry in solutions, and incorporate true sparsity using a mixed-integer nonlinear programming framework.