Section: New Results

Preconditioning of a Generalized Forward-Backward Splitting and Application to Optimization on Graphs

Participant : Loïc Landrieu [correspondent] .

Collaboration with Hugo Raguet.

In [41] , we present a preconditioning of a generalized forward-backward splitting algorithm for finding a zero of a sum of maximally monotone operators ${\sum }_{i=1}^n{A}_i+B$ with $B$ cocoercive, involving only the computation of $B$ and of the resolvent of each $A_i$ separately. This allows in particular to minimize functionals of the form ${\sum }_{i=1}^n{g}_i+f$ with $f$ smooth, using only the gradient of $f$ and the proximity operator of each $g_i$ separately. By adapting the underlying metric, such preconditioning can serve two practical purposes: first, it might accelerate the convergence, or second, it might simplify the computation of the resolvent of $A_i$ for some $i$. In addition, in many cases of interest, our preconditioning strategy allows the economy of storage and computation concerning some auxiliary variables. In particular, we show how this approach can handle large-scale, non-smooth, convex optimization problems structured on graphs, which arises in many image processing or learning applications, and that it compares favourably to alternatives in the literature.