I am giving a talk at Monash ACM seminar series on Friday, 10th of March, 2-3pm in Room 340.
Title: Open problems in convex geometry
Abstract: A convex model is the second best thing after a closed-form solution. Convex optimisation problems are often highly tractable, with a variety of numerical methods producing reliable approximations or exact solutions. The choice of the algorithms is vast, and includes general techniques such as subgradient descent or alternating projections, as well as highly specialised simplex and interior point methods. The major factors in the choice of the particular technique are the structure of the problem and the trade-offs between the resources available, the desired accuracy and the reliability of solutions.
Despite the recent breakthroughs fuelled by the addition of computational algebraic techniques to the convex optimiser's toolbox, there are deep mathematical challenges that affect the performance of numerical solvers. The major open questions which originate from the complexity analysis of numerical methods and modelling issues can be distilled into purely mathematical geometric problems, such as the polynomial Hirsch conjecture, generalised Lax conjecture and Smale's 9th problem.
In this talk I will give an overview of the aforementioned challenges and introduce some new results related to the structure of convex sets. This last part is based on my recent collaboration with Levent Tuncel (University of Waterloo), Tian Sang (RMIT University) and David Yost (Federation University Australia).