AN ALTERNATING DIRECTION METHOD OF MULTIPLIERS WITH THE CONDITIONAL GRADIENT TOTAL VARIATION METHOD FOR LINEAR INVERSE PROBLEMS

Abstract In this paper, we study the ill-posed problem using total variation regularization. To solve such a problem, we use an alternating direction method of multipliers to split our problem into two sub-problems. The novelty of our paper is in the use of the conditional gradient total variation method (CGTV) we have recently introduced. The second split- ting sub-problem is solved by transforming the obtained optimization problem to a general Sylvester matrix equation and then an orthogonal projection method is used to solve the obtained matrix equation. We give proof of the convergence of this method. Some numerical examples and applications to image restoration are given to illustrate the effectiveness of the proposed method.