Solving graph equipartition SDPs on an algebraic variety

TL;DR

This paper extends non-convex factorization methods to solve the SDP relaxation of graph equipartition problems with additional spectral constraints, enabling efficient and globally optimal solutions.

Contribution

It generalizes smooth non-convex factorization techniques to non-smooth cases involving algebraic variety constraints, with theoretical guarantees.

Findings

Efficient solution of graph equipartition SDPs with spectral bounds

Conditions under which non-convex solutions are globally optimal

Use of Riemannian optimization and augmented Lagrangian methods

Abstract

Semidefinite programs are generally challenging to solve due to their high dimensionality. Burer and Monteiro developed a non-convex approach to solve linear SDP problems by applying its low rank property. Their approach is fast because they used factorization to reduce the problem size. In this paper, we focus on solving the SDP relaxation of a graph equipartition problem, which involves an additional semidefinite upper bound constraint over the traditional linear SDP. By applying the factorization approach, we get a non-convex problem with an additional non-smooth spectral inequality constraint. We discuss when the non-convex problem is equivalent to the original SDP, and when a second order stationary point of the non-convex problem is also a global minimum. Our results generalize previous works on smooth non-convex factorization approaches for linear SDP to the non-smooth case. Moreover, the constraints of the non-convex problem involve an algebraic variety with some conducive properties that allow us to use Riemannian optimization techniques and non-convex augmented Lagrangian method to solve the SDP problem very efficiently with certified global optimality.