SIPO: Structured nonconvex optimisation: theory, algorithms and applications

Abstract:
Optimisation lies at the foundation of applied mathematics, providing the quantitative framework for selecting an optimal decision from a set of candidate solutions. For decades, the field has primarily relied on convex models, yielding clean theoretical convergence guarantees and reliable numerical algorithms. However, modern machine learning applications frequently give rise to loss functions that are intrinsically nonconvex. In these settings, traditional convex paradigms fall short, requiring novel algorithmic strategies tailored to nonconvex problems.

In this talk, we explore optimisation methods designed for nonconvex problems that explicitly exploit the underlying structure of the models. In particular, we will cover splitting methods, a family of algorithms that decompose complex models into simpler, computationally tractable subproblems. We will also discuss the central theoretical and practical difficulties inherent in solving nonconvex problems. These include the necessity of settling for stationary points or local minimisers rather than global optima, as well as pathological convergence issues, such as conservative stepsizes that may stall the iterative process. Finally, we will present recent developments aimed at overcoming these bottlenecks, illustrating how structure-exploiting algorithms bridge the gap between rigorous mathematical theory and practical numerical performance.

Speaker: Felipe Atenas (CMM, U. de Chile)