Analysis of the Adaptive Iterative Bregman Algorithm

In this paper we introduce and analyze the Adaptive Iterative Bregman algorithm, which can be viewed as a variation of other known Augmented Lagrangian Methods for the solution of constrained optimization problems of the type min v∈H J(v) subject to Av = f, where J is a convex, proper, and lower semicontinuous functional on a Hilbert spaceH and Av = f is a linear constraint. The algorithm alternates a proximity map iteration, based on forward-backward splitting, and the iterative update of a suitable Lagrange multiplier to enforce the linear constraint. We can show that, at the cost of performing a small and adaptive number of inner proximity map iterations, we can gain extra properties for the proposed algorithm, very desirable for concrete applications: in particular the execution of the iterations is made simple by forward-backward splitting, the discrepancy functional v → ‖Av − f‖ is monotone when evaluated on the iterations, and eventually we have guaranteed convergence to a solution of the given optimization problem. AMS subject classification: 65K10, 52A41, 49M30, 68U10