Semidefinite optimization is a subfield of convex optimization that generalizes linear and second-order cone programming. While semidefinite optimization problems arise in a diverse set of fields (including machine learning, robotics, and finance), efficiently solving them remains an active area of research. Developing methods that detect and exploit useful structure---such as symmetry, sparsity, or degeneracy---is one research topic. Such methods include facial and symmetry reduction, which have been successful in several applications, often reducing solve time by orders of magnitude. Nevertheless, theoretical and practical barriers preclude their general purpose use: to our knowledge, no solver uses facial or symmetry reduction as an automatic preprocessing step. This thesis addresses some of these barriers in three parts: the first develops more practical facial reduction techniques, the second proposes a more powerful and computationally efficient generalization of symmetry reduction (which we call Jordan reduction), and the third specializes techniques to semidefinite relaxations of polynomial optimization problems. We also present computational results and provide software implementations.
Thesis Supervisors: Pablo Parrilo and Russ Tedrake