Convex algebraic geometry and semidefinite optimization

Abstract

In the past decade there has been a surge of interest in algebraic approaches to optimization problems defined by multivariate polynomials. Fundamental mathematical challenges that arise in this area include understanding the structure of nonnegative polynomials, the interplay between efficiency and complexity of different representations of algebraic sets, and the development of effective algorithms. Remarkably, and perhaps unexpectedly, convexity provides a new viewpoint and a powerful framework for addressing these questions. This naturally brings us to the intersection of algebraic geometry, optimization, and convex geometry, with an emphasis on algorithms and computation. This emerging area has become known as convex algebraic geometry.This tutorial will focus on basic and recent developments in convex algebraic geometry, and the associated computational methods based on semidefinite programming for optimization problems involving polynomial equations and inequalities. There has been much recent progress, by combining theoretical results in real algebraic geometry with semidefinite programming to develop effective computational approaches to these problems. We will make particular emphasis on sum of squares decompositions, general duality properties, infeasibility certificates, approximation/inapproximability results, as well as survey the many exciting developments that have taken place in the last few years.