Journal
JOURNAL OF SCIENTIFIC COMPUTING
Volume 76, Issue 3, Pages 1698-1717Publisher
SPRINGER/PLENUM PUBLISHERS
DOI: 10.1007/s10915-018-0680-3
Keywords
Fixed-point iteration; Nonexpansive operator; Chambolle-Pock; Primal-dual; Three-operator splitting
Categories
Funding
- NSF [DMS-1621798]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1621798] Funding Source: National Science Foundation
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In this paper, we propose a new primal-dual algorithm for minimizing , where f, g, and h are proper lower semi-continuous convex functions, f is differentiable with a Lipschitz continuous gradient, and is a bounded linear operator. The proposed algorithm has some famous primal-dual algorithms for minimizing the sum of two functions as special cases. E.g., it reduces to the Chambolle-Pock algorithm when and the proximal alternating predictor-corrector when . For the general convex case, we prove the convergence of this new algorithm in terms of the distance to a fixed point by showing that the iteration is a nonexpansive operator. In addition, we prove the O(1 / k) ergodic convergence rate in the primal-dual gap. With additional assumptions, we derive the linear convergence rate in terms of the distance to the fixed point. Comparing to other primal-dual algorithms for solving the same problem, this algorithm extends the range of acceptable parameters to ensure its convergence and has a smaller per-iteration cost. The numerical experiments show the efficiency of this algorithm.
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