Variational analysis has come of age. Long an elegant theoretical toolkit for variational mathematics and nonsmooth optimization, it now increasingly underpins the study of algorithms, and a rich interplay with semi-algebraic geometry illuminates its generic applicability. As an example, alternating projections - a rudimentary but enduring algorithm for exploring the intersection of two arbitrary closed sets - concisely illustrates several far-reaching and interdependent variational ideas. A transversality measure, intuitively an angle and generically nonzero, controls several key properties: the method's linear convergence rate, a posteriori error bounds, sensitivity to data perturbations, and robustness relative to problem description. These linked ideas emerge in a wide variety of computational problems. Optimization in particular is rich in examples that depend, around critical points, on “active" manifolds of nearby approximately critical points. Such manifolds, central to classical theoretical and computational optimization, exist generically in the semi-algebraic case. We discuss examples from eigenvalue optimization and stable polynomials in control systems, and a prox-linear algorithm for large-scale composite optimization applications such as machine learning.
(This lecture is a replay of an invited talk at last year’s International Congress of Mathematicians in Seoul.)