Mathematical Optimization for Data Science Group, Saarland University

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

P. Ochs

Abstract:
A local convergence result for an abstract descent method 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 forward-backward splitting method: iPiano (a generalization of the Heavy-ball method). Moreover, it reveals an equivalence between iPiano and inertial averaged/alternating proximal minimization and projection methods. Key for this equivalence is the attraction to a local minimum within a 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.



pdf	Bibtex	Publisher's link	View-pdf	arXiv

Citation:
P. Ochs:
Local Convergence of the Heavy-ball Method and iPiano for Non-convex Optimization.

Journal of Optimization Theory and Applications, 177(1):153-180, 2018.

Bibtex:
@article{Ochs18, title = {Local Convergence of the Heavy-ball Method and iPiano for Non-convex Optimization}, author = {P. Ochs}, year = {2018}, journal = {Journal of Optimization Theory and Applications}, number = {1}, volume = {177}, pages = {153--180} }

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