TIME-VARYING SEMIDEFINITE PROGRAMMING: PATH FOLLOWING A BURER-MONTEIRO FACTORIZATION\ast

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 timevarying 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.