Alternating direction method of multipliers with variable metric indefinite proximal terms for convex optimization

Yan Gu; Nobuo Yamashita

doi:10.3934/naco.2020047

Article Contents

2020, Volume 10, Issue 4: 487-510. Doi: 10.3934/naco.2020047

This issue Previous Article A trust region algorithm for computing extreme eigenvalues of tensors Next Article Parameter-related projection-based iterative algorithm for a kind of generalized positive semidefinite least squares problem

Alternating direction method of multipliers with variable metric indefinite proximal terms for convex optimization

Yan Gu^1, , and
Nobuo Yamashita^1,

1.
Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan

^* Corresponding author: Yan Gu
^* Corresponding author: Yan Gu

This paper is dedicated in honor and to the memory of Professor Wenyu Sun

Received: May 2020

Revised: September 2020

Published: September 2020

This work is supported by JSPS KAKENHI Grant Number 17K00032

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Abstract

This paper studies a proximal alternating direction method of multipliers (ADMM) with variable metric indefinite proximal terms for linearly constrained convex optimization problems. The proximal ADMM plays an important role in many application areas, since the subproblems of the method are easy to solve. Recently, it is reported that the proximal ADMM with a certain fixed indefinite proximal term is faster than that with a positive semidefinite term, and still has the global convergence property. On the other hand, Gu and Yamashita studied a variable metric semidefinite proximal ADMM whose proximal term is generated by the BFGS update. They reported that a slightly indefinite matrix also makes the algorithm work well in their numerical experiments. Motivated by this fact, we consider a variable metric indefinite proximal ADMM, and give sufficient conditions on the proximal terms for the global convergence. Moreover, based on the BFGS update, we propose a new indefinite proximal term which can satisfy the conditions for the global convergence. Experiments on several datasets demonstrated that our proposed variable metric indefinite proximal ADMM outperforms most of the comparison proximal ADMMs.

Keywords:

Alternating direction method of multipliers,
Variable metric indefinite proximal term,
BFGS update,
Global convergence,
Convex optimization.

Mathematics Subject Classification: Primary: 90C25, 90C53, 65K05.

Citation:

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Figure 1. Evolution of the objective function values with respect to iterations on real datasets

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Figure 2. Evolution of the objective function values with respect to iterations on synthetic datasets

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Figure 3. Comparison on the CPU time

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Table 1. Comparisons among existing methods

method	$ S, T $	$ \alpha $	$ \tau $
Classical ADMM [9,11]	$ S=T=0 $	$ (0, (1+\sqrt 5)/2) $	-
Semi-proximal ADMM [7]	fixed positive semidefinite matrices	$ (0, (1+\sqrt 5)/2) $	$ (1, +\infty) $
Variable semi-proximal ADMM [13,14]	variable positive semidefinite matrices	1	$ (1, +\infty) $
Indefinite proximal ADMM [20]	fixed indefinite matrices	1	$ (0.75,1) $
Proposed method	variable indefinite matrices	1	$ (0.75,1) $

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Construction of $ T_k $ via the BFGS update

1 Let $ \delta \in (0, \infty) $, $ \tau \in (\frac3 4, 1) $ and $ B_0 \succeq \tau M $;
2 Let $ c_k $ be a sequence such that $ c_k \in [0,1] $ and $ \sum\limits_{k=0}^{\infty} c_k < \infty $;
3 If $ s_k \neq 0 $, then set $ \tilde{l}_k = M^{\delta} s_k $ and update $ B_{k+1} $ via
$\begin{equation*} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B_{k+1} = B_{k}+ c_k \left(\frac {{\tilde{l}}_{k}{\tilde{l}}_{k}^\top}{{\tilde{l}}_{k}^\top {s}_{k}} -{\frac {B_{k}{s}_{k}{s}_{k}^\top B_{k}^\top}{{s}_{k}^\top B_{k}{s}_{k}}} \right); \end{equation*} $
4 Otherwise
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B_{k+1} = B_k;$
5 Construct $ T_{k+1} $ as
$\begin{equation*} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ T_{k+1} = B_{k+1} - M. \end{equation*} $

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Table 2. Results for real datasets

Datasets	Tolerance	$ \beta $	ADMM		SPADM		IPADM		IADMB		IADML
Datasets	Tolerance	$ \beta $	Iter.	T-A	Iter.	T-A	Iter.	T-A	Iter.	T-A	Iter.	T-A
Boston-house $ m=506 $ $ n=13 $	$ \epsilon^ \mathrm{rel} = 10^{-2} $	100	22.0	0.0010	70.0	0.0015	48.0	0.0013	26.0	0.0015	25.0	0.0018
	$ \epsilon ^ \mathrm{abs} = 10^{-3} $	300	9.0	0.0010	37.0	0.0013	32.0	0.0012	20.0	0.0015	21.0	0.0016
		500	11.0	0.0010	29.0	0.0012	26.0	0.0011	20.0	0.0012	19.0	0.0014
	$ \epsilon ^ \mathrm{abs} = 10^{-3} $	300	16.0	0.0010	69.0	0.0014	60.0	0.0013	29.0	0.0014	{34.0}	0.0017
	$ \epsilon ^ \mathrm{abs} = 10^{-4} $	500	18.0	0.0011	75.0	0.0015	65.0	0.0014	28.0	0.0015	30.0	0.0016
		700	25.0	0.0010	86.0	0.0017	75.0	0.0015	36.0	0.0017	36.0	0.0017

California-housing $ m=20640 $ $ n=8 $	$ \epsilon^ \mathrm{rel} = 10^{-2} $	1000	36.0	0.0014	89.0	0.0069	77.0	0.0064	38.0	0.0022	38.0	0.0054
	$ \epsilon ^ \mathrm{abs} = 10^{-3} $	1500	24.0	0.0012	73.0	0.0057	64.0	0.0056	26.0	0.0022	27.0	0.0050
		1700	21.0	0.0013	67.0	0.0056	59.0	0.0049	24.0	0.0018	24.0	0.0041
	$ \epsilon ^ \mathrm{abs} = 10^{-3} $	5000	23.0	0.0013	55.0	0.0046	49.0	0.0044	28.0	0.0020	28.0	0.0048
	$ \epsilon ^ \mathrm{abs} = 10^{-4} $	5300	22.0	0.0014	53.0	0.0045	47.0	0.0043	27.0	0.0023	27.0	0.0042
		5500	22.0	0.0012	52.0	0.0041	47.0	0.0042	26.0	0.0018	26.0	0.0036

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Table 3. Results for Synthetic datasets ($ \epsilon^ \mathrm{rel} = 10^{-2} $, $ \epsilon ^ \mathrm{abs} = 10^{-3} $)

Data	T-LU/ME	$ \beta $	ADMM		SPADM		IPADM		PADML		IADML
Data	T-LU/ME	$ \beta $	Iter.	T-A	Iter.	T-A	Iter.	T-A	Iter.	T-A	Iter.	T-A
$ m=2664 $	T-LU$ =10.315 $	300	30.7	0.5069	75.2	0.5898	66.9	0.5286	36.3	0.2987	34.1	0.2784
$ n=2778 $	T-ME$ =1.980 $	500	17.9	0.3010	49.9	0.4102	42.5	0.3555	26.5	0.2260	24.5	0.2034
		800	11.5	0.1888	36.0	0.2854	31.7	0.2539	20.1	0.1695	18.3	0.1507
		1000	10.3	0.1716	31.6	0.2537	27.3	0.2232	18.4	0.1578	17.1	0.1437

$ m=3580 $	T-LU$ =28.516 $	300	44.9	1.4417	111.5	1.9481	95.9	1.6807	50.2	0.9149	47.8	0.8656
$ n=4620 $	T-ME$ =5.378 $	500	27.9	0.9129	73.1	1.2882	62.1	1.0911	36.0	0.6608	33.7	0.6198
		800	17.9	0.5888	53.2	0.9468	42.5	0.7578	27.9	0.5178	25.4	0.4786
		1000	14.3	0.4787	41.1	0.7307	35.1	0.6266	24.6	0.4576	22.9	0.4221

$ m=7498 $	T-LU=118.826	500	46.8	1.7680	86.9	4.0866	67.2	3.1514	47.2	2.2569	47.0	2.2453
$ n=5633 $	T-ME=16.141	1000	24.3	0.9105	51.0	2.3636	45.0	2.0981	27.6	1.3902	26.1	1.2590
		1500	16.1	0.6303	40.9	1.9333	36.0	1.6943	20.7	0.9949	19.5	0.9493
		2000	12.1	0.4829	28.9	1.3641	25.9	1.2279	18.8	0.9107	17.0	0.8338

$ m=4774 $	T-LU=158.214	500	48.0	4.7976	134.3	8.7370	116.7	7.6785	61.6	4.1815	59.7	3.9039
$ n=10368 $	T-ME=24.249	1000	23.4	2.3597	80.7	5.3301	70.8	5.7369	41.6	3.6014	38.1	2.6846
		1500	17.4	1.4561	55.7	3.0283	49.0	2.6745	31.4	1.7645	29.1	1.6197
		2000	13.7	1.1631	47.0	2.6077	41.2	2.2898	29.1	1.6362	26.3	1.4693

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References

[1]	S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Foundations and Trends® in Machine Learning, 3 (2011), 1–122. doi: 10.1561/2200000016.
[2]	A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, Journal of Mathematical Imaging and Vision, 40 (2011), 120-145. doi: 10.1007/s10851-010-0251-1.
[3]	W. Deng and W. Yin, On the global and linear convergence of the generalized alternating direction method of multipliers, Journal of Scientific Computing, 66 (2016), 889-916. doi: 10.1007/s10915-015-0048-x.
[4]	J. Eckstein, Some saddle-function splitting methods for convex programming, Optimization Methods and Software, 4 (1994), 75-83. doi: 10.1080/10556789408805578.
[5]	J. Eckstein and D. P. Bertsekas, On the douglas-rachford splitting method and the proximal point algorithm for maximal monotone operators, Mathematical Programming, 55 (1992), 293-318. doi: 10.1007/BF01581204.
[6]	J. Eckstein and W. Yao, Relative-error approximate versions of Douglas–Rachford splitting and special cases of the ADMM, Mathematical Programming, 170 (2018), 417-444. doi: 10.1007/s10107-017-1160-5.
[7]	M. Fazel, T. K. Pong, D. Sun and P. Tseng, Hankel matrix rank minimization with applications to system identification and realization, SIAM Journal on Matrix Analysis and Applications, 34 (2013), 946-977. doi: 10.1137/110853996.
[8]	M. Fortin and R. Glowinski, Chapter ⅲ on decomposition-coordination methods using an augmented lagrangian, in Studies in Mathematics and Its Applications, vol. 15, Elsevier, (1983), 97–146. doi: 10.1016/S0168-2024(08)70028-6.
[9]	D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximation, Computers and Mathematics with Applications, 2 (1976), 17-40. doi: 10.1016/0898-1221(76)90003-1.
[10]	B. Gao and F. Ma, Symmetric alternating direction method with indefinite proximal regularization for linearly constrained convex optimization, Journal of Optimization Theory and Applications, 176 (2018), 178-204. doi: 10.1007/s10957-017-1207-z.
[11]	R. Glowinski and A. Marroco, Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité d'une classe de problèmes de dirichlet non linéaires, ESAIM: Mathematical Modelling and Numerical Analysis-Modélisation Mathématique et Analyse Numérique, 9 (1975), 41–76. doi: 10.1051/m2an/197509R200411.
[12]	M. L. N. Gonçalves, M. M. Alves and J. G. Melo, Pointwise and ergodic convergence rates of a variable metric proximal alternating direction method of multipliers, Journal of Optimization Theory and Applications, 177 (2018), 448-478. doi: 10.1007/s10957-018-1232-6.
[13]	Y. Gu and N. Yamashita, An alternating direction method of multipliers with the BFGS update for structured convex quadratic optimization, preprint, arXiv: 1903.02270.
[14]	Y. Gu and N. Yamashita, A proximal ADMM with the Broyden family for convex optimization problems, Journal of Industrial and Management Optimization, 13 (2020). doi: 10.3934/jimo.2020091.
[15]	D. Harrison Jr and D. L. Rubinfeld, Hedonic housing prices and the demand for clean air, Journal of Environmental Economics and Management, 5 (1978), 81-102. doi: 10.1016/0095-0696(78)90006-2.
[16]	B. He, L.-Z. Liao, D. Han and H. Yang, A new inexact alternating directions method for monotone variational inequalities, Mathematical Programming, 92 (2002), 103-118. doi: 10.1007/s101070100280.
[17]	B. He, H. Liu, Z. Wang and X. Yuan, A strictly contractive Peaceman–Rachford splitting method for convex programming, SIAM Journal on Optimization, 24 (2014), 1011-1040. doi: 10.1137/13090849X.
[18]	B. He and X. Yuan, On the $O(1/n)$ convergence rate of the Douglas-Rachford alternating direction method, SIAM Journal on Numerical Analysis, 50 (2012), 700-709. doi: 10.1137/110836936.
[19]	B. He and X. Yuan, Block-wise alternating direction method of multipliers for multiple-block convex programming and beyond, SMAI Journal of Computational Mathematics, 1 (2015), 145-174. doi: 10.5802/smai-jcm.6.
[20]	B. He, F. Ma and X. Yuan, Optimally linearizing the alternating direction method of multipliers for convex programming, Computational Optimization and Applications, (2019), 1–28. doi: 10.1007/s10589-019-00152-3.
[21]	K. Koh, S.-J. Kim and S. Boyd, An interior-point method for large-scale l1-regularized logistic regression, Journal of Machine Learning Research, 8 (2007), 1519-1555. doi: 10.1109/JSTSP.2007.910971.
[22]	M. Li, D. Sun and K.-C. Toh, A majorized admm with indefinite proximal terms for linearly constrained convex composite optimization, SIAM Journal on Optimization, 26 (2016), 922-950. doi: 10.1137/140999025.
[23]	P.-L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM Journal on Numerical Analysis, 16 (1979), 964-979. doi: 10.1137/0716071.
[24]	P. A. Lotito, L. A. Parente and M. Solodov, A class of variable metric decomposition methods for monotone variational inclusions, Journal of Convex Analysis, 16 (2009), 857-880.
[25]	Y. Nesterov, Gradient methods for minimizing composite functions, Mathematical Programming, 140 (2013), 125-161. doi: 10.1007/s10107-012-0629-5.
[26]	R. K. Pace and R. Barry, Sparse spatial autoregressions, Statistics and Probability Letters, 33 (1997), 291-297. doi: 10.1016/S0167-7152(96)00140-X.
[27]	L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.
[28]	H. Trevor, T. Robert and F. JH, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer Science and Business Media, 2009.
[29]	P. Tseng and S. Yun, A coordinate gradient descent method for nonsmooth separable minimization, Mathematical Programming, 117 (2009), 387-423. doi: 10.1007/s10107-007-0170-0.
[30]	M. Xu, Proximal alternating directions method for structured variational inequalities, Journal of Optimization Theory and Applications, 134 (2007), 107-117. doi: 10.1007/s10957-007-9192-2.
[31]	M. Xu and T. Wu, A class of linearized proximal alternating direction methods, Journal of Optimization Theory and Applications, 151 (2011), 321-337. doi: 10.1007/s10957-011-9876-5.
[32]	W. Yin, S. Osher, D. Goldfarb and J. Darbon, Bregman iterative algorithms for $l_1$-minimization with applications to compressed sensing, SIAM Journal on Imaging Sciences, 1 (2008), 143-168. doi: 10.1137/070703983.