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Abstract
Multi-block linear constrained separable convex minimizations are ubiquitous and have been drawing increasing attention in recent researches. The alternating direction method of multipliers (ADMM) has been well studied and used in the literature for the two-block case. However, the direct extension of the ADMM to the multi-block case is not necessarily convergent. ADMM with Gaussian Back Substitution and ADMM with Prox-Parallel Splitting are two useful schemes to deal with the multi-block situation. Nevertheless, only a sublinear convergence rate was given in previous studies. In this paper, we prove the linear convergence rate of these two schemes under some assumptions. The proofs mainly depend on the variational inequalities.
Highlights
INTRODUCTIONWe consider the multi-block constrained problem which reads as m m min θi(xi) s.t. i=1 i=1 where xi ∈ Rni , θi is convex but may be non-differentiable
In this paper, we consider the multi-block constrained problem which reads as m m min θi(xi) s.t
We focus on exploiting the linear convergence rate of these two multi-block algorithms
Summary
We consider the multi-block constrained problem which reads as m m min θi(xi) s.t. i=1 i=1 where xi ∈ Rni , θi is convex but may be non-differentiable. For the two-block case, the alternating direction method of multipliers (ADMM) [15]–[17] is an efficient algorithm and. That is to say that the scheme (3) is not convergent in the general case, where Lβ is the augmented Lagrangian function, which will be defined . X. Deng et al.: Linear Convergence Rate of Splitting Algorithms for Multi-Block Constrained Convex Minimizations. The linear convergence rate of two multi-block ADMM schemes has still not been studied. For the two multi-block ADMMs, it is quite challenging to establish the contraction property due to that complicated scheme directly. The variational inequality technique can transform their convergence analysis to simple matrix analysis, which is the fundamental method used in our paper
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