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Pure Mathematics 2020
正则化交替方向乘子算法求解非凸不可分离问题
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Abstract:
本文作者提出了一种新的正则化交替方向乘子法来解决非凸优化问题，在不要求正则项严格凸的情况下证明了算法的全局收敛性，在增广拉格朗日函数满足KL性质的条件下，证明了算法的强收敛性，并且通过应用于Lasso模型求解，证明了算法的有效性。
In this paper, we propose a new regularized alternating direction method of multipliers to solve the non-convex optimization problem. We prove the global convergence of the algorithm without requiring the strict convexity of the regular term. Under the condition that the augmented Lagrangian function satisfies KL function, the strong convergence of the algorithm is proved, and the effectiveness of the algorithm is proved by applying it to the Lasso model.
| [1] | Guo, K., Han, D. and Wu, T.T. (2017) Convergence of Alternating Direction Method for Minimizing Sum of Two Nonconvex Functions with Linear Constraints. International Journal of Computer Mathematics, 94, 1653-1669. https://doi.org/10.1080/00207160.2016.1227432 |
| [2] | Guo, K., Han, D., Wang, D.Z.W., et al. (2017) Convergence of ADMM for Multi-Block Nonconvex Separable Optimization Models. Frontiers of Mathematics in China, 12, 1139-1162. https://doi.org/10.1007/s11464-017-0631-6 |
| [3] | Li, G.Y. and Pong, T.K. (2014) Global Convergence of Splitting Methods for Nonconvex Composite Optimization. EprintArxiv, 25, 2434-2460. https://doi.org/10.1137/140998135 |
| [4] | Guo, K. and Wang, X. (2018) Convergence of Generalized Alternating Direction Method of Multipliers for Nonseparable Nonconvex Objective with Linear Constraints. Journal of Mathematical Research with Applications, 38, 87-104. |
| [5] | Chen, C.H., Li, M., Liu, X., et al. (2019) Extended ADMM and BCD for Nonseparable Convex Minimization Models with Quadratic Coupling Terms: Convergence Analysis and Insights. Mathematical Programming, 173, 37-77. https://doi.org/10.1007/s10107-017-1205-9 |
| [6] | Gao, X. and Zhang, S.Z. (2017) First-Order Algorithms for Convex Optimization with Nonseparable Objective and Coupled Constraints. Journal of the Operations Research Society of China, 5, 131-159.
https://doi.org/10.1007/s40305-016-0131-5 |
| [7] | Jian, J.B., Zhang, Y. and Chao, M.T. (2019) A Regularized Al-ternating Direction Method of Multipliers for a Class of Nonconvex Problems. Journal of Inequalities and Applications, 2019, 1-16. https://doi.org/10.1186/s13660-019-2145-0 |
| [8] | Bolte, J., Sabach, S. and Teboulle, M. (2014) Proximal Alter-nating Linearized Minimization for Nonconvex and Nonsmooth Problems. Mathematical Programming, 146, 459-494. https://doi.org/10.1007/s10107-013-0701-9 |
| [9] | Attouch, H., Bolte, J., Redont, P., et al. (2010) Proximal Alter-nating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Lojasiewicz Inequality. Mathematics of Operations Research, 35, 438-457. https://doi.org/10.1287/moor.1100.0449 |
