Academic Journal
The Proximal Alternating Direction Method of Multipliers in the Nonconvex Setting: Convergence Analysis and Rates: Convergence analysis and rates
| Title: | The Proximal Alternating Direction Method of Multipliers in the Nonconvex Setting: Convergence Analysis and Rates: Convergence analysis and rates |
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| Authors: | Radu Ioan Boţ, Dang-Khoa Nguyen |
| Source: | Mathematics of Operations Research. 45:682-712 |
| Publication Status: | Preprint |
| Publisher Information: | Institute for Operations Research and the Management Sciences (INFORMS), 2020. |
| Publication Year: | 2020 |
| Subject Terms: | 101016 Optimisation, NONSMOOTH, 0211 other engineering and technologies, Convergence rates, 02 engineering and technology, proximal splitting algorithms, convergence analysis, SPLITTING ALGORITHM, 47H05, 65K05, 90C26, Convergence analysis, Kurdyka-Lojasiewicz property, FOS: Mathematics, Mathematics - Numerical Analysis, Lojasiewicz exponent, Mathematics - Optimization and Control, Proximal splitting algorithms, SUM, alternating direction method of multipliers, Variable metric, Numerical Analysis (math.NA), Kurdyka-Łojasiewicz property, Łojasiewicz exponent, Alternating direction method of multipliers, nonconvex complexly structured optimization problems, variable metric, Optimization and Control (math.OC), convergence rates, MINIMIZATION, POINTS, 101016 Optimierung, Nonconvex complexly structured optimization problems |
| Description: | We propose two numerical algorithms in the fully nonconvex setting for the minimization of the sum of a smooth function and the composition of a nonsmooth function with a linear operator. The iterative schemes are formulated in the spirit of the proximal alternating direction method of multipliers and its linearized variant, respectively. The proximal terms are introduced via variable metrics, a fact that allows us to derive new proximal splitting algorithms for nonconvex structured optimization problems, as particular instances of the general schemes. Under mild conditions on the sequence of variable metrics and by assuming that a regularization of the associated augmented Lagrangian has the Kurdyka–Łojasiewicz property, we prove that the iterates converge to a Karush–Kuhn–Tucker point of the objective function. By assuming that the augmented Lagrangian has the Łojasiewicz property, we also derive convergence rates for both the augmented Lagrangian and the iterates. |
| Document Type: | Article |
| Language: | English |
| ISSN: | 1526-5471 0364-765X |
| DOI: | 10.1287/moor.2019.1008 |
| DOI: | 10.48550/arxiv.1801.01994 |
| Access URL: | http://arxiv.org/pdf/1801.01994 http://arxiv.org/abs/1801.01994 https://ucrisportal.univie.ac.at/de/publications/e526ce94-59b7-48e9-9406-221df3dcb56a https://doi.org/10.1287/moor.2019.1008 https://pubsonline.informs.org/doi/10.1287/moor.2019.1008 https://dblp.uni-trier.de/db/journals/mor/mor45.html#BotN20 https://ideas.repec.org/a/inm/ormoor/v45y2020i2p682-712.html http://ui.adsabs.harvard.edu/abs/2018arXiv180101994I/abstract |
| Rights: | arXiv Non-Exclusive Distribution |
| Accession Number: | edsair.doi.dedup.....b7ddf6dbd94d9512ce06445dbdea4d5a |
| Database: | OpenAIRE |
| ISSN: | 15265471 0364765X |
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| DOI: | 10.1287/moor.2019.1008 |