Convex geometry and optimisation (6 hours lectures)
Dr. Vera Roshchina (School of Mathematics and Statistics, UNSW Sydney)
Lecture 1: Polyhedral problems and Complexity
This lecture starts with an introduction to the geometry of polytopes and then focus on major open problems related to polytopes, including Smale’s 9th problem, polynomial Hirsch conjecture and related issues of complexity estimates for numerical methods, illustrating the discussion with classical algorithms.
Lecture 2: Faces of Convex Sets
In the second lecture we briefly review major definitions and results pertaining to the boundary structure of convex sets, then focus on the geometry of spectrahedra, dimensions of faces, lifts and shadows of convex sets.
Lecture 3: Conic Programming
In this lecture we will look at generic Conic programming problems, including semidefinite feasibility and optimisation problems, and key applications. In particular, applications to machine learning and Goemans-Williamson algorithm for the MAX-CUT problem will be discussed.
Lecture 4: Duality in Conic Programming
In this lecture we will briefly recall the basics of geometric duality, such as separation theorems and polars, then focus on duality in conic programming and accompanying issues, such as ill-posed pairs of feasibility problems, duality gap, strong duality and facial reduction.
Lecture 5: Polynomial Problems
This lecture will cover semidefinite programming formulations for the sums of squares representations and the generalised Lax conjecture.
Lecture 6: Facially Dual Complete Cones and Lexicographic Tangents
We will look at key geometric notions describing good and bad arrangements of faces of convex sets, focussing on the relations between them and on constructive characterisations.
Dr Vera Roshchina is a Senior Lecturer at the School of Mathematics and Statistics, UNSW Sydney. Her research interests are in geometry of finite dimensional optimisation problems. This includes subdifferential calculus, facial structure of convex sets and miscellaneous topics in real complexity and dynamical systems.
Vera received her PhD in applied mathematics from the City University of Hong Kong in 2009 under the supervision of Prof. Felipe Cucker. Prior to joining UNSW Sydney in 2018, she was an ARC DECRA Research Fellow at RMIT University and held postdoctoral positions at the The University of Melbourne, Federation University Australia and University of Évora (Portugal).
See the website http://www.roshchina.com/cv/ for more info on Dr Vera Roshchina.