Geometry and Convexity in Optimization

Abstract

Convex geometry is a fascinating area of simplicity and elegance yet mathematically rich. It is home to many beautiful and surprising theorems with numerous applications ranging from optimisation practical ones such as robotics, economics, and machine learning. In these year’s lectures, we go over the basics of facial structure of convex sets, starting with the finite dimensional setting. We review the tools that help study geometric properties of convex sets and to construct convex sets with desirable properties. We then focus on structured convex problems, predominantly those defined algebraically (through matrix and polynomial inequalities and representations). Finally, we review some properties and behaviours of convex sets that are specific to the infinite-dimensional setting. The fundamental mathematical ideas and phenomena will be contextualised in optimisation applications, including conic programming and projection methods.