  We propose a first-order augmented Lagrangian algorithm (FALC) to solve the
composite norm minimization problem min |sigma(F(X)-G)|_alpha + |C(X)- d|_beta
subject to A(X)-b in Q; where sigma(X) denotes the vector of singular values of
X, the matrix norm |sigma(X)|_alpha denotes either the Frobenius, the nuclear,
or the L2-operator norm of X, the vector norm |.|_beta denotes either the
L1-norm, L2-norm or the L infty-norm; Q is a closed convex set and A(.), C(.),
F(.) are linear operators from matrices to vector spaces of appropriate
dimensions. Basis pursuit, matrix completion, robust principal component
pursuit (PCP), and stable PCP problems are all special cases of the composite
norm minimization problem. Thus, FALC is able to solve all these problems in a
unified manner. We show that any limit point of FALC iterate sequence is an
optimal solution of the composite norm minimization problem. We also show that
for all epsilon > 0, the FALC iterates are epsilon-feasible and epsilon-optimal
after O(log(1/epsilon)) iterations, which require O(1/epsilon) constrained
shrinkage operations and Euclidean projection onto the set Q. Surprisingly, on
the problem sets we tested, FALC required only O(log(1/epsilon)) constrained
shrinkage, instead of the O(1/epsilon) worst case bound, to compute an
epsilon-feasible and epsilon-optimal solution. To best of our knowledge, FALC
is the first algorithm with a known complexity bound that solves the stable PCP
problem.
