ORIE Colloquium: Jérôme Bolte (Toulouse)

An O-Minimal Optimization Perspective

This talk explores o-minimal optimization, a framework for treating a wide class of structured problems, including piecewise linear, semi-algebraic, and more generally, problems defined by simple analytic formulas. This makes it particularly well-suited to modern computational languages such as Python. After giving an overview of what o-minimal structures are and why they are fundamental for optimization, we will focus on a simple but versatile tool called the projection formula established in collaboration with Daniilidis, Lewis, and Shiota. This formula bridges two views on nonsmoothness: for geometers, nonsmooth functions are smooth once the domain is partitioned into manifolds; for optimizers, this is not necessary, since subdifferentials already carry the right amount of first-order information. We will show how the projection formula yields powerful results, including nonsmooth Sard-type theorems, a nonsmooth Lojasiewicz inequality, a nonsmooth chain rule, and a formal subdifferential framework known as conservative calculus. These results, in turn, have notable algorithmic implications for proximal-gradient and Lagrangian-like methods, stochastic subgradient methods, and automatic differentiation.

Bio: Jerome Bolte is a full professor at the Toulouse School of Economics and holds a Chair in Artificial Intelligence at the Artificial Intelligence Institute of Toulouse (ANITI). He studied pure and applied mathematics before completing a degree in mathematics and then a doctorate at Montpellier University. Prior to moving to Toulouse in 2010, he spent six years as an associate professor at Sorbonne University in Paris and one year at Ecole Polytechnique. He received the SIAM Optimization Prize in 2017, along with S. Sabach and M. Teboulle, for work at the crossroads of semi-algebraic geometry and first-order methods. He is currently an associate editor at Mathematical Programming and Foundations of Computational Mathematics. His research interests range from continuous optimization to machine learning.