- Fast Multiple Splitting Algorithms for Convex Optimization Donald Goldfarb (goldfarbcolumbia.edu) Shiqian Ma (sm2756columbia.edu) Abstract: We present in this paper two different classes of general $K$-splitting algorithms for solving finite-dimensional convex optimization problems. Under the assumption that the function being minimized has a Lipschitz continuous gradient, we prove that the number of iterations needed by the first class of algorithms to obtain an $\epsilon$-optimal solution is $O(1/\epsilon)$. The algorithms in the second class are accelerated versions of those in the first class, where the complexity result is improved to $O(1/\sqrt{\epsilon})$ while the computational effort required at each iteration is almost unchanged. To the best of our knowledge, the complexity results presented in this paper are the first ones of this type that have been given for splitting and alternating direction type methods. Moreover, all algorithms proposed in this paper are parallelizable, which makes them particularly attractive for solving certain large-scale problems. Keywords: Convex Optimization, Variable Splitting, Alternating Direction Method, Alternating Linearization Method, Complexity Theory, Decomposition, Smoothing Techniques, Parallel Computing, Proximal Point Algorithm, Optimal Gradient Method Category 1: Convex and Nonsmooth Optimization Citation: Technical Report, Department of IEOR, Columbia University, 2009 Download: [PDF]Entry Submitted: 12/22/2009Entry Accepted: 12/22/2009Entry Last Modified: 03/18/2011Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Programming Society.