© 2016 American Automatic Control Council (AACC). In this paper, we consider the model reduction problem of large-scale systems, such as systems obtained through the discretization of partial differential equations. We propose a computationally optimal randomized proper orthogonal decomposition (RPOD∗) technique to obtain the reduced order model by perturbing the primal and adjoint system using Gaussian white noise. We show that the computations required by the RPOD∗ algorithm is orders of magnitude cheaper when compared to the balanced proper orthogonal decomposition (BPOD) algorithm while the performance of the RPOD∗ algorithm is better than BPOD. It is optimal in the sense that a minimal number of snapshots is needed. We also relate the RPOD∗ algorithm to random projection algorithms. One numerical example is given to illustrate the procedure.