Hyperbolicity Cones and Sparse Optimization

Hyperbolicity cones are convex cones that appear in the intersection of convex optimization and real algebraic geometry. There are many examples of such cones which are useful in optimization, such as the cone of positive semidefinite matrices. This talk will introduce the concept of hyperbolicity cones and describe new constructions of hyperbolicity cones which generalize a well-known construction called the Renegar derivative. We will also describe how these new hyperbolicity cones relate to another type of convex cone which encodes sparse optimization problems such as sparse linear regression and sparse PCA. We will give some concrete algorithmic applications as well as tight bounds on the eigenvalues of matrices where all small principal submatrices are positive semidefinite.