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A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science. (English) Zbl 1206.90117
Summary: We generalize the primal-dual hybrid gradient (PDHG) algorithm proposed by {\it M. Zhu} and {\it T. F. Chan} in [“An efficient primal-dual hybrid gradient algorithm for total variation image restoration”, CAM Report 08--34, UCLA, Los Angeles, CA (2008)] to a broader class of convex optimization problems. In addition, we survey several closely related methods and explain the connections to PDHG. We point out convergence results for a modified version of PDHG that has a similarly good empirical convergence rate for total variation (TV) minimization problems. We also prove a convergence result for PDHG applied to TV denoising with some restrictions on the PDHG step size parameters. We show how to interpret this special case as a projected averaged gradient method applied to the dual functional. We discuss the range of parameters for which these methods can be shown to converge. We also present some numerical comparisons of these algorithms applied to TV denoising, TV deblurring, and constrained $l_1$ minimization problems.

##### MSC:
 90C25 Convex programming 90C06 Large-scale problems (mathematical programming) 49K35 Minimax problems (optimality conditions) 49N45 Inverse problems in calculus of variations 65K10 Optimization techniques (numerical methods)
NESTA; RecPF
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