Linearized block-wise alternating direction method of multipliers for multiple-block convex programming

Zhongming Wu; Xingju Cai; Deren Han

doi:10.3934/jimo.2017078

Article Contents

2018, Volume 14, Issue 3: 833-855. Doi: 10.3934/jimo.2017078

This issue Previous Article Next Article Ergodic control for a mean reverting inventory model

Linearized block-wise alternating direction method of multipliers for multiple-block convex programming

1.
School of Economics and Management, Southeast University, Nanjing 210096, China
2.
School of Mathematical Sciences, Jiangsu Key Labratory for NSLSCS, Nanjing Normal University, Nanjing 210023, China

1 Corresponding author
1 Corresponding author

Received: May 31, 2016

Revised: July 31, 2017

Early access: August 2017

Published: June 2018

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Abstract

The alternating direction method of multipliers (ADMM) is an efficient approach for two-block separable convex programming, while it is not necessarily convergent when extending this method to multiple-block case directly. One appealing method is that converts the multiple-block variables into two groups firstly and then adopts the classic ADMM with inexact solving to the resulting model, which is so-called block-wise ADMM. However, solving the subproblems in block-wise ADMM are usually difficult when the linear mappings in the constraints are not diagonal or the proximal operator of the objective function is not easy to evaluate. Therefore, in this paper, we adopt the linearization technique to different terms presented in the block-wise ADMM subproblems, and obtain three kinds of linearized block-wise ADMM which make the subproblems easy to solve in general case. Moreover, under some mild conditions, we prove the global convergence of the three new methods and report some preliminary numerical results to indicate the feasibility and effectiveness of the linearization strategy.

Keywords:

Convex programming,
separable structure,
multiple-block,
alternating direction method of multipliers,
linearization technique,
global convergence.

Mathematics Subject Classification: Primary: 90C25, 65K05, 49M27.

Citation:

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Figure 1. Test images. (a) 256 × 256 House, (b) 256 × 256 Boy, (c) 512 × 512 Barbara, (d) 512 × 512 Tomandjerry

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Figure 2. The case of $A=I$: decomposed cartoons (above row) and textures (below row) by Algorithm 1

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Figure 3. Evolutions of objective function values with respect to iterations and computing time for the case of $A=I$ in the first 50 iterations (test image: Boy image)

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Figure 4. Evolutions of objective function values with respect to iterations and computing time for the case of $A=S$ (test image: Boy image)

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Figure 5. The case of $A=S$: test images with missing pixel (1st column), decomposed cartoon (2nd column) and textures (3rd column) by Algorithm 1, reconstructed images (4th column) by superposing the corresponding cartoons and textures

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Table 1. Numerical results for quadratic programming problem (4.1)

$n_1$	$n_2=l$	$n_3$	Algorithm H			Algorithm 1			Algorithm 2			Algorithm 3
			Iter.	Initer.	Time	Iter.	Initer.	Time	Iter.	Initer.	Time	Iter.	Time
100	100	100	71.0	7848.0	0.34	50.0	1661.0	0.08	162.0	825.0	0.09	69.0	0.04
200	200	200	85.0	12015.0	0.70	62.0	2316.0	0.16	255.0	1685.0	0.39	85.0	0.05
200	300	200	84.0	9790.0	0.83	51.0	1487.0	0.14	327.0	2311.0	0.69	88.0	0.05
300	300	300	93.0	16177.0	1.55	53.0	1527.0	0.22	348.0	2784.0	1.20	92.0	0.11
500	300	500	103.0	30080.0	5.77	56.0	2842.0	0.98	441.0	3528.0	3.39	85.0	0.59
500	500	500	90.0	21683.0	4.52	60.0	2390.0	1.02	507.0	4161.0	4.81	92.0	0.66
500	600	500	91.0	21828.0	5.23	59.0	2178.0	1.22	548.0	4615.0	5.97	93.0	0.70
600	600	600	93.0	25832.0	10.16	61.0	2637.0	2.06	710.0	6390.0	10.16	90.0	1.20
600	800	600	91.0	28772.0	13.72	62.0	2255.0	2.56	692.0	6228.0	11.84	91.0	1.31
800	800	800	93.0	224730.0	270.98	64.0	2185.0	5.73	801.0	7209.0	25.84	91.0	3.11
800	1000	800	98.0	235200.0	292.06	64.0	2237.0	6.25	1134.0	10218.0	40.56	105.0	3.25
1000	1000	1000	108.0	255521.0	484.45	64.0	2033.0	10.70	1226.0	11065.0	62.58	195.0	7.16

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