The integration of mathematical optimization with machine learning represents a significant advancement in addressing complex learning tasks. While many machine learning models rely on optimization for training, the use of tailored optimization techniques has shown particular promise in handling non-convex formulations. This interdisciplinary approach has led to substantial improvements in solving challenging machine learning problems, particularly those involving unsupervised or semi-supervised learning paradigms.
The topic of this talk reinforces the synergy between optimization and machine learning, showing how global optimization techniques can enhance model quality and robustness. Specifically, we explore two fundamental problems: the Minimum Sum of Squares clustering problem (MSSC), including its constrained or semi-supervised variant, and Semi-Supervised Support Vector Machines (S3VM). MSSC aims to partition a set of data points into a fixed number of clusters while minimizing the sum of Euclidean distances to cluster centroids. The integration of background knowledge transforms the MSSC into a semi-supervised task. Similarly, S3VM extends traditional SVMs by leveraging both labeled and unlabeled data to learn a more reliable decision boundary.
The complexity of these machine learning tasks makes the associated optimization problems NP-hard, necessitating advanced optimization techniques for efficient solutions. By employing exact algorithms and semidefinite programming tools, we achieve significant advancements in scalability and solution quality with respect to state-of-the-art methods. Through comprehensive analysis, we demonstrate the efficacy of global solutions in enhancing model performance and generalization.
The outcome of this research underscores the cross-fertilization between mathematical optimization and machine learning, demonstrating the mutual benefits when combining expertise from both domains.

This research has been jointly conducted by Veronica Piccialli, Anna Russo Russo, Jan Schwiddessen, Antonio M. Sudoso, and Angelika Wiegele.