Solving saddle point problems: a landscape of primal-dual algorithm with larger stepsizes

Fan Jiang (), Zhiyuan Zhang () and Hongjin He ()
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Fan Jiang: Nanjing University of Information Science and Technology
Zhiyuan Zhang: Xiamen University
Hongjin He: Ningbo University

Journal of Global Optimization, 2023, vol. 85, issue 4, No 2, 846 pages

Abstract: Abstract We consider a class of saddle point problems frequently arising in the areas of image processing and machine learning. In this paper, we propose a simple primal-dual algorithm, which embeds a general proximal term induced with a positive definite matrix into one subproblem. It is remarkable that our algorithm enjoys larger stepsizes than many existing state-of-the-art primal-dual-like algorithms due to our relaxed convergence-guaranteeing condition. Moreover, our algorithm includes the well-known primal-dual hybrid gradient method as its special case, while it is also of possible benefit to deriving partially linearized primal-dual algorithms. Finally, we show that our algorithm is able to deal with multi-block separable saddle point problems. In particular, an application to a multi-block separable minimization problem with linear constraints yields a parallel algorithm. Some computational results sufficiently support the promising improvement brought by our relaxed requirement.

Keywords: Saddle point problem; Primal-dual algorithm; Composite optimization; Convex programming; Image processing; RPCA (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10898-022-01233-0

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