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

TL;DR

This paper introduces a novel framework called Mixed-Projection Conic Optimization for modeling and solving low-rank problems with certifiable optimality, leveraging symmetric projection matrices and advanced algorithms.

Contribution

It presents a new modeling paradigm using symmetric projection matrices for low-rank constraints, along with algorithms that solve these problems to certifiable optimality.

Findings

Successfully solves low-rank problems to certifiable optimality.

Scales algorithms to matrices with up to 600 dimensions.

Outperforms existing heuristics and relaxations.

Abstract

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 tractable convex optimization problems over the non-convex 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, for the first time, solve low-rank optimization problems to certifiable optimality. Using currently available spatial branch-and-bound codes, not tailored to projection matrices, we can scale our exact (resp. near-exact) algorithms to matrices with up to 30 (resp. 600) rows/columns. Our algorithms also supply certifiably near-optimal solutions for larger problem sizes and outperform existing heuristics, by deriving an alternative to the popular nuclear norm relaxation which generalizes the perspective relaxation from vectors to matrices. 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.