A unified approach for minimizing composite norms

We propose a first-order augmented Lagrangian algorithm (FALC) to solve the composite norm minimization problem min XeRmn denotes the vector of singular values of XeR × n}, the matrix norm σ (X) a denotes either the Frobenius, the nuclear, or the l₂ -operator norm of X, the vector norm . β denotes either the l₁ -norm, l₂ -norm or the l-norm; mathcalQ is a closed convex set and A, C F are linear operators from R m×n 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 e >0, the FALC iterates are e-feasible and e-optimal after O log (e-1) iterations, which require {O}(e-1) constrained shrinkage operations and Euclidean projection onto the set {Q}. Surprisingly, on the problem sets we tested, FALC required only \mathcal{O }(\log (e-1) constrained shrinkage, instead of the Oe-1 worst case bound, to compute an e-feasible and e-optimal solution. To best of our knowledge, FALC is the first algorithm with a known complexity bound that solves the stable PCP problem.