Norm Optimization Via Alternating Direction Method Of Multipliers

The alternating direction method of multipliers(ADMM), a simple but efficient algorithm that is well suited to distributed convex optimization, and particularly to large-scale problem-s. With the advantage of a decomposition-coordination procedure, the solutions to small local subproblems are easier to obtained and in coordinated to find a solution to a large global prob-lem. Many problems of recent interest in several fields can be posed in the framework of convex optimization, such as l1-minimization in Compressive Sensing,TV-minimization in Image Pro-cessing,Matrix Fitting Models,Minimization on Stiefel Manifold. ADMM can be viewed as an effective method to solve those problems above. It is considered to discuss further on matrix function problems while vector related function with affine constraint are already well solved.This article mainly discusses solving F-norm and Ky Fan k-norm minimization problems with affine and semi-definite constraints via two kinds of slightly changed ADMM. We first turn F-norm primal problem and Ky Fan k-norm primal problem into general pattern of ADMM, then compute the subproblem via projection onto closed convex set in order to get solution(under the condition A is onto); when the resulting subproblems do not have closed-form solution, we discuss and solve the subproblems via Linearized ADMM. And particular, we focus on the two special value of k, which transform our Ky Fan k-norm optimization into nuclear norm and spectral norm problems,respectively. We demonstrate those two optimization problems can be solved effectively via the Linearized ADMM. Finally, We demonstrate the effectiveness of ADMM, including giving some numerical examples and test data.