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A first-order primal-dual algorithm for convex problems with applications to imaging. (English) Zbl 1255.68217
Summary: In this paper we study a first-order primal-dual algorithm for non-smooth convex optimization problems with known saddle-point structure. We prove convergence to a saddle-point with rate $O\left(1/N\right)$ in finite dimensions for the complete class of problems. We further show accelerations of the proposed algorithm to yield improved rates on problems with some degree of smoothness. In particular we show that we can achieve $O\left(1/{N}^{2}\right)$ convergence on problems, where the primal or the dual objective is uniformly convex, and we can show linear convergence, i.e. $O\left({\omega }^{N}\right)$ for some $\omega \in \left(0,1\right)$, on smooth problems. The wide applicability of the proposed algorithm is demonstrated on several imaging problems such as image denoising, image deconvolution, image inpainting, motion estimation and multi-label image segmentation.

##### MSC:
 68U10 Image processing (computing aspects) 90C25 Convex programming 94A08 Image processing (compression, reconstruction, etc.)