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

In this talk, a local convergence result for abstract descent methods in
non-convex optimization is presented. In particular, the analysis is
tailored to inertial methods. The result can be summarized as follows:
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. Moreover, it reveals an equivalence between iPiano (a
generalization of the Heavy-ball method) and inertial averaged/alternating proximal
minimization and projection methods. Key for this equivalence is the
attraction to a local minimum within a common neighborhood and the fact that, for a prox-regular function, the gradient of the Moreau envelope
is locally Lipschitz continuous and expressible in terms of the proximal
mapping. In a numerical feasibility problem, the inertial alternating
projection method significantly outperforms its non-inertial variants.

Joint CIA-CCIMI seminar