  We present a novel approach to maximizing the transmission rate in a MIMO
relay system, where all nodes are equipped with multiple antennas and the relay
is self-sustained by harvesting energy. We formulate an optimization problem
and use dual-characterization to derive a closed-form solution for the optimal
power splitting ratio and precoding design. We propose an efficient primal-dual
algorithm to jointly optimize the power allocation at source and relay for
transmission and the power splitting at relay for energy harvesting, and show
that using non-uniform power splitting is optimal. Numerical results
demonstrate the significant rate gain of non-uniform power splitting over
traditional uniform splitting especially at low source transmit power. We also
analyze our algorithm numerically and demonstrate its efficiency at reducing
the run-time by several orders of magnitudes compared to a standard solver,
\textcolor{blue}{and existing algorithms in literature
