The Interplay of Discrete Optimization and Machine Learning

Modern society increasingly relies on optimization and artificial intelligence to make complex decisions in areas such as manufacturing, transportation, energy, healthcare, and scientific discovery. As these technologies become more widely deployed, there is a growing need to understand how they can work together effectively and, equally importantly, when their predictions and recommendations can be trusted. This award supports research that seeks to develop the mathematical foundations for integrating machine learning with optimization algorithms, enabling future decision-making systems that are more efficient and reliable. By establishing rigorous theoretical guarantees for learning-enhanced optimization methods, this research has the potential to improve the performance and trustworthiness of computational tools used across a broad range of engineering and scientific applications. The project will also educate graduate and undergraduate students at the intersection of optimization, artificial intelligence, and applied mathematics through research, coursework, and interdisciplinary collaborations. This research aims to investigate the fundamental interplay between mixed-integer optimization, discrete geometry, and machine learning. The research develops new mathematical tools to determine how much data is needed for learning-enhanced optimization algorithms to perform reliably and to characterize the expressive power and computational complexity of neural network models used in optimization. At the same time, formal methods from optimization and discrete geometry will be harnessed to answer foundational questions about machine learning itself, creating a two-way exchange of ideas between the two fields. The resulting algorithms, mathematical frameworks, and theoretical guarantees are expected to advance both optimization and artificial intelligence by establishing a rigorous foundation for future learning-enhanced decision-making methods. 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.