Mixed-Projection Conic Optimization: A New Paradigm for Modeling Rank Constraints

We propose a framework for modeling and solving low-rank optimization problems to certifiable optimality. We introduce symmetric projection matrices that satisfy Y-2 = Y, the matrix analog of binary variables that satisfy z(2) = z, to model rank constraints. By leveraging regularization and strong duality, we prove that this modeling paradigm yields convex optimization problems over the nonconvex set of orthogonal projection matrices. Furthermore, we design outer-approximation algorithms to solve low-rank problems to certifiable optimality, compute lower bounds via their semidefinite relaxations, and provide near optimal solutions through rounding and local search techniques. We implement these numerical ingredients and, to our knowledge, for the first time solve low-rank optimization problems to certifiable optimality. Our algorithms also supply certifiably near-optimal solutions for larger problemsizes and outperformexisting heuristics by deriving an alternative to the popular nuclear norm relaxation. Using currently available spatial branch-and-bound codes, not tailored to projection matrices, we can scale our exact (respectively, near-exact) algorithms to matrices with up to 30 (600) rows/columns. All in all, our framework, which we name mixed-projection conic optimization, solves low-rank problems to certifiable optimality in a tractable and unified fashion.