Primal-dual block-proximal splitting for a class of non-convex problems. ETNA - Electronic Transactions on Numerical Analysis

Résumé 0

We develop block structure-adapted primal-dual algorithms for non-convexnon-smooth optimisation problems, whose objectives can be written as compositions$G(x)+F(K(x))$ of non-smooth block-separable convex functions $G$ and $F$ with anonlinear Lipschitz-differentiable operator $K$. Our methods are refinements ofthe nonlinear primal-dual proximal splitting method for such problems withoutthe block structure, which itself is based on the primal-dual proximal splittingmethod of Chambolle and Pock for convex problems. We propose individual steplength parameters and acceleration rules for each of the primal and dual blocksof the problem. This allows them to convergence faster by adapting to thestructure of the problem. For the squared distance of the iterates to a criticalpoint, we show local $O(1/N)$, $O(1/N^2)$, and linear rates under varyingconditions and choices of the step length parameters. Finally, we demonstrate the performance of the methods for the practical inverseproblems of diffusion tensor imaging and electrical impedance tomography.