  We consider the estimation of the state transition matrix in vector
autoregressive models, when time sequence data is limited but nonsequence
steady-state data is abundant. To leverage both sources of data, we formulate
the least squares minimization problem regularized by a Lyapunov penalty. We
impose cardinality or rank constraints to reduce the complexity of the
autoregressive model. We solve the resulting nonconvex, nonsmooth problem by
using the proximal alternating linearization method (PALM). We show that PALM
is globally convergent to a critical point and that the estimation error
monotonically decreases. Furthermore, we obtain explicit formulas for the
proximal operators to facilitate the implementation of PALM. We demonstrate the
effectiveness of the developed method on synthetic and real-world data. Our
experiments show that PALM outperforms the gradient projection method in both
computational efficiency and solution quality.
