  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.
