Unifying abstract inexact convergence theorems and block coordinate variable metric iPiano

An abstract convergence theorem for a class of generalized descent methods that explicitly models relative errors is proved. The convergence theorem generalizes and unifies several recent abstract convergence theorems. It is applicable to possibly non-smooth and non-convex lower semi-continuous functions that satisfy the Kurdyka–Lojasiewicz (KL) inequality, which comprises a huge class of problems. Most of … Read more

Local Convergence of the Heavy-ball Method and iPiano for Non-convex Optimization

A local convergence result for abstract descent methods is proved. The sequence of iterates is attracted by a local (or global) minimum, stays in its neighborhood and converges within this neighborhood. This result allows algorithms to exploit local properties of the objective function. In particular, the abstract theory in this paper applies to the inertial … Read more