Article Contents
Article Contents

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

• 1 Corresponding author
• 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.

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

 Citation:

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

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

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)

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

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

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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