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On the Convergence of Block Coordinate Descent Type Methods
Abstract
In this paper we study smooth convex programming problems where the decision variables vector is split into several blocks of variables. We analyze the block coordinate gradient projection method in which each iteration consists of performing a gradient projection step with respect to a certain block taken in a cyclic order. Global sublinear rate of convergence of this method is established and it is shown that it can be accelerated when the problem is unconstrained. In the unconstrained setting we also prove a sublinear rate of convergence result for the so-called alternating minimization method when the number of blocks is two. When the objective function is also assumed to be strongly convex, linear rate of convergence is established.
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Cites
A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems
Beck, A., & Teboulle, M. (2009) · SIAM Journal on Imaging Sciences · doi:10.1137/080716542 · Cited by 11,952
Self-citationBertsekas, D. P. (1997) · Journal of the Operational Research Society · doi:10.1038/sj.jors.2600425 · Cited by 11,726
Iterative Solution of Nonlinear Equations in Several Variables
Ortega, J. M., & Rheinboldt, W. C. (2000) · Society for Industrial and Applied Mathematics eBooks · doi:10.1137/1.9780898719468 · Cited by 6,860
Introductory Lectures on Convex Optimization
Nesterov, Y. (2004) · Applied Optimization · doi:10.1007/978-1-4419-8853-9 · Cited by 2,698
A method for solving the convex programming problem with convergence rate O(1/k^2)
Nesterov, Y. E. (1983) · Доклады Академии наук · Cited by 2,662
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Efficiency of Coordinate Descent Methods on Huge-Scale Optimization Problems
Nesterov, Y. (2012) · SIAM Journal on Optimization · doi:10.1137/100802001 · Cited by 1,187
Parallel and distributed computation
Bertsekas, D. P., Tsitsiklis, J. N., & Sennikov, A. N. (1989) · Cited by 1,187
Gradient methods for minimizing composite objective function
Nesterov, Y. E. (2007) · Cited by 867
Richtárik, P., & Takáč, M. (2014) · Mathematical Programming · doi:10.1007/s10107-012-0614-z · Cited by 654
Cited by
Wright, S. J. (2015) · Mathematical Programming · doi:10.1007/s10107-015-0892-3 · Cited by 1,438
Hong, M., Razaviyayn, M., Luo, Z.-Q., & Pang, J.-S. (2016) · IEEE Signal Processing Magazine · doi:10.1109/msp.2015.2481563 · Cited by 447
Robust Low-Rank Tensor Recovery: Models and Algorithms
Goldfarb, D., & Qin, Z. (. (2014) · SIAM Journal on Matrix Analysis and Applications · doi:10.1137/130905010 · Cited by 408
A Globally Convergent Algorithm for Nonconvex Optimization Based on Block Coordinate Update
Xu, Y., & Yin, W. (2017) · Journal of Scientific Computing · doi:10.1007/s10915-017-0376-0 · Cited by 285
Metadata
- Authors
- Beck, A., & Tetruashvili, L.
- Venue
- SIAM Journal on Optimization
- Published
- 2013-01
- Volume
- 23
- Issue
- 4
- Pages
- 2037-2060
- ISSN
- 1052-6234 (ISSN-L)
- 1095-7189
- OpenAlex
- W2023901033
- Cited by
- 572
- References
- 16
- FWCI
- 34.60
- Type
- journal article
- Language
- en
- Open access
- closed
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