  In this paper, we formulate the interference alignment (IA) problem for a
multiuser multiple-input multiple-output (MIMO) system in the presence of an
eavesdropper as a rank constrained rank minimization (RCRM) problem. The aim of
the proposed rank minimization IA schemes is to find the precoding and receiver
subspace matrices to align interference and wiretapped signals into the lowest
dimension subspaces while keeping the desired signal subspace spanning full
available spatial dimensions. To deal with the nonconvexity of the rank
function, we present two convex relaxations of the RCRM problem, namely nuclear
norm (NN) and reweighted nuclear norm (RNN), and transform the rank constraints
to equivalent and tractable ones. We then derive a coordinate decent approach
to obtain the solutions for IA schemes. The simulation results show that our
proposed IA designs outperform the conventional IA design in terms of average
secrecy sum rate. On the other hand, our proposed designs perform the same or
better than other secure IA schemes which account for low interference and
wiretapped signal power rather than for low dimensions of interference and
wiretapped signal matrices in the systems which achieve the perfect IA.
