Abstract
In this study, we introduce a primal-dual prediction-correction algorithm framework for convex optimization problems with known saddle-point structure. Our unified frame adds the proximal term with a positive definite weighting matrix. Moreover, different proximal parameters in the frame can derive some existing well-known algorithms and yield a class of new primal-dual schemes. We prove the convergence of the proposed frame from the perspective of proximal point algorithm-like contraction methods and variational inequalities approach. The convergence rate O(1/t) in the ergodic and nonergodic senses is also given, where t denotes the iteration number.
Highlights
We consider the following model that arises from various signal and image processing applications: min x f (Bx) + f (x), ( )where B is a continuous linear operator, and f and f are proper convex lower-semicontinuous functions
Bonettini and Ruggiero [ ] established the convergence of a general primal-dual method for nonsmooth convex optimization problems and showed that the convergence of the scheme can be considered as an -subgradient method on the primal formulation of the variational problem when the steplength parameters are a priori selected sequences
In Section, we propose the primal-dualbased contraction algorithm framework in prediction-correction fashion
Summary
We consider the following model that arises from various signal and image processing applications: min x f (Bx). Bonettini and Ruggiero [ ] established the convergence of a general primal-dual method for nonsmooth convex optimization problems and showed that the convergence of the scheme can be considered as an -subgradient method on the primal formulation of the variational problem when the steplength parameters are a priori selected sequences. He and Yuan [ ] did a novel study on these primal-dual algorithms from the perspective of contraction perspective. Zhang, Zhu, and Wang [ ] proposed a simple primal-dual method for total-variation image restoration problems and showed that their iterative scheme has the O( /k) convergence rate in the ergodic sense.
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