iPiano: inertial proximal algorithm for nonconvex optimization. In this paper we study an algorithm for solving a minimization problem composed of a differentiable (possibly nonconvex) and a convex (possibly nondifferentiable) function. The algorithm iPiano combines forward-backward splitting with an inertial force. It can be seen as a nonsmooth split version of the Heavy-ball method from Polyak. A rigorous analysis of the algorithm for the proposed class of problems yields global convergence of the function values and the arguments. This makes the algorithm robust for usage on nonconvex problems. The convergence result is obtained based on the Kurdyka-Łojasiewicz inequality. This is a very weak restriction, which was used to prove convergence for several other gradient methods. First, an abstract convergence theorem for a generic algorithm is proved, and then iPiano is shown to satisfy the requirements of this theorem. Furthermore, a convergence rate is established for the general problem class. We demonstrate iPiano on computer vision problems – image denoising with learned priors and diffusion based image compression

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  1. Benning, Martin; Riis, Erlend Skaldehaug; Schönlieb, Carola-Bibiane: Bregman Itoh-Abe methods for sparse optimisation (2020)
  2. Gao, Xue; Cai, Xingju; Han, Deren: A Gauss-Seidel type inertial proximal alternating linearized minimization for a class of nonconvex optimization problems (2020)
  3. Kang, Myeongmin: Approximate versions of proximal iteratively reweighted algorithms including an extended IP-ICMM for signal and image processing problems (2020)
  4. Liu, Wenchen; Tang, Yincai; Wu, Xianyi: Separating variables to accelerate non-convex regularized optimization (2020)
  5. Banert, Sebastian; Bot, Radu Ioan: A general double-proximal gradient algorithm for d.c. programming (2019)
  6. Clason, Christian; Mazurenko, Stanislav; Valkonen, Tuomo: Acceleration and global convergence of a first-order primal-dual method for nonconvex problems (2019)
  7. de Oliveira, Welington; Tcheou, Michel P.: An inertial algorithm for DC programming (2019)
  8. Li, Y.; Sixou, B.; Peyrin, F.: Nonconvex mixed TV/Cahn-Hilliard functional for super-resolution/segmentation of 3D trabecular bone images (2019)
  9. Lu, Jian; Yang, Hanmei; Shen, Lixin; Zou, Yuru: Ultrasound image restoration based on a learned dictionary and a higher-order MRF (2019)
  10. Ochs, Peter: Unifying abstract inexact convergence theorems and block coordinate variable metric iPiano (2019)
  11. Tan, Pauline; Pierre, Fabien; Nikolova, Mila: Inertial alternating generalized forward-backward splitting for image colorization (2019)
  12. Tovey, Robert; Benning, Martin; Brune, Christoph; Lagerwerf, Marinus J.; Collins, Sean M.; Leary, Rowan K.; Midgley, Paul A.; Schönlieb, Carola-Bibiane: Directional sinogram inpainting for limited angle tomography (2019)
  13. Valkonen, Tuomo: Block-proximal methods with spatially adapted acceleration (2019)
  14. Wu, Zhongming; Li, Min: General inertial proximal gradient method for a class of nonconvex nonsmooth optimization problems (2019)
  15. Bednarczuk, E. M.; Jezierska, A.; Rutkowski, K. E.: Proximal primal-dual best approximation algorithm with memory (2018)
  16. Benning, Martin; Burger, Martin: Modern regularization methods for inverse problems (2018)
  17. Boţ, Radu Ioan; Csetnek, Ernö Robert; Nimana, Nimit: An inertial proximal-gradient penalization scheme for constrained convex optimization problems (2018)
  18. Cazelles, Elsa; Seguy, Vivien; Bigot, Jérémie; Cuturi, Marco; Papadakis, Nicolas: Geodesic PCA versus log-PCA of histograms in the Wasserstein space (2018)
  19. Geiping, Jonas; Moeller, Michael: Composite optimization by nonconvex majorization-minimization (2018)
  20. Iutzeler, Franck; Malick, Jérôme: On the proximal gradient algorithm with alternated inertia (2018)

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