Block coordinate update (BCU) methods enjoy low per-update computational complexity because every time only one or a few block variables would need to be updated among possibly a large number of blocks. They are also easily parallelized and thus have been particularly popular for solving problems involving large-scale dataset and/or variables. In this paper, we propose a primaldual BCU method for solving linearly constrained convex program with multi-block variables. The method is an accelerated version of a primaldual algorithm proposed by the authors, which applies randomization in selecting block variables to update and establishes an <i>O</i>(1/<i>t</i>) convergence rate under convexity assumption. We show that the rate can be accelerated to O ( 1 / t 2 ) if the objective is strongly convex. In addition, if one block variable is independent of the others in the objective, we then show that the algorithm can

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Higher-order low-rank tensors naturally arise in many applications including hyperspectral data recovery, video inpainting, seismic data recon-struction, and so on. We propose a new model to recover a low-rank tensor by simultaneously performing low-rank matrix factorizations to the all-mode ma-tricizations of the underlying tensor. An alternating minimization algorithm is applied to solve the model, along with two adaptive rank-adjusting strategies when the exact rank is not known.

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We observe that Sturm’s error bounds readily imply that for semidefinite feasibility problems, the method of alternating projections converges at a rate of O(k^{-1/(2^{d+1}-2)}), where d is the singularity degree of the problem—the minimal number of facial reduction iterations needed to induce Slater’s condition. Consequently, for almost all such problems (in the sense of Lebesgue measure), alternating projections converge at a worst-case rate of O(1/k^{0.5}).