Time-Varying Semidefinite Programming: Path Following a Burer-Monteiro Factorization

TL;DR

This paper introduces an online path-following algorithm for time-varying semidefinite programs using low-rank Burer-Monteiro factorization, enabling efficient tracking of solutions over time.

Contribution

It develops a predictor-corrector method with a horizontal space constraint for TV-SDPs, improving solution tracking efficiency and accuracy.

Findings

Method maintains solutions close to the optimal path if well-initialized.

Numerical experiments show faster runtime compared to interior point methods.

Effective for max-cut SDP relaxations in dynamic settings.

Abstract

We present an online algorithm for time-varying semidefinite programs (TV-SDPs), based on the tracking of the solution trajectory of a low-rank matrix factorization, also known as the Burer-Monteiro factorization, in a path-following procedure. There, a predictor-corrector algorithm solves a sequence of linearized systems. This requires the introduction of a horizontal space constraint to ensure the local injectivity of the low-rank factorization. The method produces a sequence of approximate solutions for the original TV-SDP problem, for which we show that they stay close to the optimal solution path if properly initialized. Numerical experiments for a time-varying max-cut SDP relaxation demonstrate the computational advantages of the proposed method for tracking TV-SDPs in terms of runtime compared to off-the-shelf interior point methods.