Reduction methods in semidefinite and conic optimization

Abstract

Conic optimization, or cone programming, is a subfield of convex optimization that includes linear, second-order cone, and semidefinite programming as special cases. While conic 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 convex relaxations of polynomial optimization problems. Throughout, we place emphasis on semidefinite programs and, more generally, optimization problems over symmetric cones. We also present computational results.