CAM Colloquium, October 26, 2012 - Osman Güler

Osman Guler Department of Mathematics and Statistics University of Maryland, Baltimore County Friday, October 26, 2012 Interior-Point Methods and their Broad Impact on Sciences In 1990s, ipms started to have significant impact on various scientific disciplines. The first and very significant event was the emergence of semidefinite programming. In addition to providing powerful algorithms for solving optimization problems coming from various fields, ipms started to make connections with pure and applied mathematics (Euclidean Jordan algebras, homogeneous convex cones, Siegel domains, hyperbolic polynomials, real algebraic geometry, Einstein manifolds, Monge-Ampere equations, moment problems, etc). The theory of hyperbolic polynomials helped with the resolution of the 50-year old Lax conjecture, provided a simpler proof of the van derWaerden conjecture, but also introduced some new, hard-looking conjectures (generalized Lax conjectures). In recent times, hyperbolic polynomials have been making unexpected connections with surprisingly many fields: stability theory, graph theory, matroids, combinatorics, tropical geometry, etc. Another theoretically significant development is the recent proof (Hildebrand, Fox) that a certain Monge-Ampere equation is a universal self-concordant barrier function, with some attractive properties. In my talk, I will survey some of these developments. The optimization problems coming from some recent applications are huge. First-order methods, which have relatively low memory requirements and complexity rates that do not depend on dimension, seem well-suited for such applications. I will conclude my talk by mentioning some of these methods.