Alternating Directions Method For Constrained L1-Norm Minimization

In this thesis, we propose a couple of alternating directions methods for solving-norm constrained and l2-norm constrained l1-norm minimization problems, respectively. We show that the proposed algorithms converge globally. Moreover, we use both al-gorihtms to recover sparse signal in compressive sensing which show that the proposed algorithms are promising.In the first chapter, we give some primary results of the compressive sensing and sparse optimization, and list the recent progress of the algorithms in sparse optimization. Moreover, some important notations and symbols which used in the context are included.It is will known that when observed data contains impulse noise, the l1-norm in-equality constrained l1-norm minimization is perfect to recover sparse signal. In chapter two, we propose an alternating directions method to solve this model. Using linearized technique, adding proximal points term, and then taking full use of the l1-norm’s char-acter to make orthogonal projection, we show that each subproblem admits closed-form solutions. With some mild conditions, we establish the convergence theorem of the pro-posed algorithm. Finally, we do numerical experiments to use the proposed algorithm to recover large-scale sparse signals. The numerical results illustrate that the proposed algorithm is efficient and is competitive with the well-known solver-YALL1.We know that the l2-norm inequality constrained l1-norm is very suitable for recov-ering large-scale sparse signal when its observed data is corrupted by Guassian noise. In chapter three, we propose an alternating directions method to solve this problem. Com-paring with the highly related method-YALL1, our proposed algorithm takes full use of the constraints’structure, and it does not require linearized technique to ensure that each subproblem has closed-form solutions.In chapter4, we give a summary of this thesis and list some further research topics.