AN ELEMENTARY PROOF OF CONVERGENCE FOR THE FORWARD-BACKWARD SPLITTING ALGORITHM

作者：Byrne Charles L^*

来源：Journal of Nonlinear and Convex Analysis, 2014, 15(4): 681-691.

• Forward-backward splitting algorithm
• proximity operator
• iterative optimization

The forward-backward splitting (FBS) algorithm is a quite general iterative method that includes, as particular cases, the projected gradient descent algorithm for constrained minimization, the CQ algorithm for the split feasibility problem, the projected Landweber algorithm for constrained least squares, and the simultaneous orthogonal projection algorithm for the convex feasibility problem. The FBS algorithm involves iterating with respect to an averaged operator that is the product of two firmly non-expansive operators, one of which is Moreau%26apos;s proximity operator. The usual proof of convergence employs the Krasnosel%26apos;skii-Mann Theorem. This proof depends, therefore, on knowing that if the gradient of a convex differentiable function is non-expansive, then it is firmly non-expansive, and that the composition of averaged operators is again averaged. Neither of these results is trivial to prove, especially the former. In this paper we give an elementary proof of convergence of the FBS algorithm that does not rely on these results.

• 出版日期2014