A unified approach for minimizing composite norms

Necdet S. Aybat, G. Iyengar

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

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.

Original languageEnglish (US)
Pages (from-to)181-226
Number of pages46
JournalMathematical Programming
Volume144
Issue number1-2
DOIs
StatePublished - Jan 1 2014

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Composite
Norm
Pursuit
Composite materials
Principal Components
Minimization Problem
Vector spaces
Shrinkage
Denote
Iterate
Mathematical operators
Optimal Solution
Augmented Lagrangians
Basis Pursuit
Matrix Completion
Matrix Norm
Augmented Lagrangian
Operator Norm
L1-norm
Limit Point

All Science Journal Classification (ASJC) codes

  • Software
  • Mathematics(all)

Cite this

Aybat, Necdet S. ; Iyengar, G. / A unified approach for minimizing composite norms. In: Mathematical Programming. 2014 ; Vol. 144, No. 1-2. pp. 181-226.
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A unified approach for minimizing composite norms. / Aybat, Necdet S.; Iyengar, G.

In: Mathematical Programming, Vol. 144, No. 1-2, 01.01.2014, p. 181-226.

Research output: Contribution to journalArticle

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