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March  2021, 17(2): 971-979. doi: 10.3934/jimo.2020007

An efficient low complexity algorithm for box-constrained weighted maximin dispersion problem

Zi Xu , , Siwen Wang and  Jinjin Huang

Department of Mathematics, College of Sciences, Shanghai University, 200444 Shanghai, China

* Corresponding author: Zi Xu

Received  September 2018 Revised  August 2019 Published  March 2021 Early access  January 2020

Fund Project: The first author is supported by National Natural Science Foundation of China under the grant 11571221 and 11771208

The box-constrained weighted maximin dispersion problem is to find a point in an $ n $-dimensional box such that the minimum of the weighted Euclidean distance from given $ m $ points is maximized. In this paper, we first propose a two-phase method to solve it. In the first phase, we adopt a block successive upper bound minimization (BSUM) algorithm framework and choose a special piecewise linear upper bound function for the weighted maximin dispersion problem. The per-iteration complexity of our algorithm is very low, since the subproblem is a one-dimensional piecewise linear minimax problem over the box constraints, or eqivalently, a two-dimensional linear programming problem which can be solved in at most $ O(m) $ time by existing algorithms. In the second phase, a useful rounding is employed to enhance the solution. Moreover, we propose another strengthened two-phase algorithm, which employs a maximum improvement successive upper-bound minimization (MISUM) algorithm instead of BSUM algorithm in the first phase. At each step, only the block that provides the maximum improvement of the upper bound function is updated. Then, it can be proved that every limit point of the iterate generated by this strengthened algorithm is a stationary point. Numerical results show that the proposed algorithms are efficient.

Keywords: Maximin dispersion problem, block successive upper bound minimization, maximum improvement successive upper-bound minimization, successive convex approximation.
Mathematics Subject Classification: Primary: 90C06, 90C26; Secondary: 93E10.
Citation: Zi Xu, Siwen Wang, Jinjin Huang. An efficient low complexity algorithm for box-constrained weighted maximin dispersion problem. Journal of Industrial and Management Optimization, 2021, 17 (2) : 971-979. doi: 10.3934/jimo.2020007
References:
[1]

B. Chen, S. He, Z. Li and S. Zhang, Maximum block improvement and polynomial optimization, SIAM J. Optim., 22 (2012), 87-107.  doi: 10.1137/110834524.

[2]

B. Dasarthy and L. White, A maximin location problem, Oper. Res., 28 (1980), 1385-1401.  doi: 10.1287/opre.28.6.1385.

[3]

M. Grant and S. Boyd, CVX User's guide: For CVX version 1.21, User's Guide, (2010), 24–75.

[4]

S. Haines, J. Loeppky, P. Tseng and X. Wang, Convex relaxations of the weighted maxmin dispersion problem, SIAM J. Optim., 23 (2013), 2264-2294.  doi: 10.1137/120888880.

[5]

M. Johbson, L. Moore and D. Ylvisaker, Maximin Distance Designs, Statist. Plann. J., 26 (1990), 131-148.  doi: 10.1016/0378-3758(90)90122-B.

[6]

A. Konar and N. Sidiropoulos, Fast approximation algorithms for a class of nonconvex QCQP problems using first-order methods, IEEE Trans. Signal Process, 65 (2017), 3494-3509.  doi: 10.1109/TSP.2017.2690386.

[7]

N. Megiddo, Linear-time algorithms for linear programming in $\mathbb{R}^3$ and related problems, SIAM J. Comput., 12 (1983), 759-776.  doi: 10.1137/0212052.

[8]

R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ. 1970.

[9]

S. Ravi, D. Rosenkrantz and G. Tayi, Heuristic and special case algorithms for dispersion problems, Oper. Res., 42 (1994), 299-310.  doi: 10.1287/opre.42.2.299.

[10]

M. Razaviyayn, M. Hong and Z.-Q. Luo, A unified convergence analysis of block successive minimization methods for nonsmooth optimization, SIAM J. Optim., 23 (2013), 1126-1153.  doi: 10.1137/120891009.

[11]

S. Wang and Y. Xia, On the ball-constrained weighted maximin dispersion problem, SIAM J. Optim., 26 (2016), 1565-1588.  doi: 10.1137/15M1047167.

[12]

D. J. White, A heuristic approach to a weighted maxmin disperation problem, IMA J. Math. Appl. Bus. Indust., 7 (1996), 219-231.  doi: 10.1093/imaman/7.3.219.

[13]

Z. Wu, Y. Xia and S. Wang, Approximating the weighted maximin dispersion problem over an $\ell_{p}-$ball: SDP relaxation is misleading, Optim. Lett., 12 (2018), 875-883.  doi: 10.1007/s11590-017-1177-y.

show all references

References:
[1]

B. Chen, S. He, Z. Li and S. Zhang, Maximum block improvement and polynomial optimization, SIAM J. Optim., 22 (2012), 87-107.  doi: 10.1137/110834524.

[2]

B. Dasarthy and L. White, A maximin location problem, Oper. Res., 28 (1980), 1385-1401.  doi: 10.1287/opre.28.6.1385.

[3]

M. Grant and S. Boyd, CVX User's guide: For CVX version 1.21, User's Guide, (2010), 24–75.

[4]

S. Haines, J. Loeppky, P. Tseng and X. Wang, Convex relaxations of the weighted maxmin dispersion problem, SIAM J. Optim., 23 (2013), 2264-2294.  doi: 10.1137/120888880.

[5]

M. Johbson, L. Moore and D. Ylvisaker, Maximin Distance Designs, Statist. Plann. J., 26 (1990), 131-148.  doi: 10.1016/0378-3758(90)90122-B.

[6]

A. Konar and N. Sidiropoulos, Fast approximation algorithms for a class of nonconvex QCQP problems using first-order methods, IEEE Trans. Signal Process, 65 (2017), 3494-3509.  doi: 10.1109/TSP.2017.2690386.

[7]

N. Megiddo, Linear-time algorithms for linear programming in $\mathbb{R}^3$ and related problems, SIAM J. Comput., 12 (1983), 759-776.  doi: 10.1137/0212052.

[8]

R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ. 1970.

[9]

S. Ravi, D. Rosenkrantz and G. Tayi, Heuristic and special case algorithms for dispersion problems, Oper. Res., 42 (1994), 299-310.  doi: 10.1287/opre.42.2.299.

[10]

M. Razaviyayn, M. Hong and Z.-Q. Luo, A unified convergence analysis of block successive minimization methods for nonsmooth optimization, SIAM J. Optim., 23 (2013), 1126-1153.  doi: 10.1137/120891009.

[11]

S. Wang and Y. Xia, On the ball-constrained weighted maximin dispersion problem, SIAM J. Optim., 26 (2016), 1565-1588.  doi: 10.1137/15M1047167.

[12]

D. J. White, A heuristic approach to a weighted maxmin disperation problem, IMA J. Math. Appl. Bus. Indust., 7 (1996), 219-231.  doi: 10.1093/imaman/7.3.219.

[13]

Z. Wu, Y. Xia and S. Wang, Approximating the weighted maximin dispersion problem over an $\ell_{p}-$ball: SDP relaxation is misleading, Optim. Lett., 12 (2018), 875-883.  doi: 10.1007/s11590-017-1177-y.

Table 1.  Numerical results for Algorithm 2, Algorithm 4 and a randomized algorithm in [13]
$ n $ $ m $ CR Algorithm in [13] Algorithm 2 Algorithm 4
max min ave time(s) $ f(\bar{x}) $ $ f(\tilde{x}) $ time(s) $ f(\bar{x}) $ $ f(\tilde{x}) $ time(s)
10 5 27.23 24.46 9.12 15.41 0.17 25.26 24.46 0.30 25.06 24.62 0.11
10 18.84 17.24 5.53 10.38 0.16 15.75 15.73 0.07 16.65 15.99 0.08
15 20.70 18.58 4.88 10.12 0.19 16.98 16.98 0.10 18.23 17.68 0.16
20 19.61 16.58 4.27 9.07 0.19 12.76 12.30 0.08 14.91 12.75 0.24
50 25 125.01 100.04 64.56 82.91 0.36 100.11 98.81 0.65 100.83 97.12 1.45
50 119.37 97.73 58.83 78.78 0.46 102.54 102.52 0.51 96.28 96.24 1.93
75 116.26 91.54 61.52 77.06 0.65 93.45 91.85 0.56 93.50 92.08 2.00
100 111.27 90.98 57.09 74.20 0.80 87.20 85.38 0.56 90.22 88.04 2.40
100 50 248.61 203.88 149.11 179.52 0.64 205.35 203.69 1.11 206.61 206.25 5.57
100 237.30 201.19 148.41 174.23 0.91 194.10 198.58 1.14 198.03 195.88 5.19
150 229.82 189.37 143.86 169.15 1.60 190.54 189.85 1.17 191.68 191.02 6.89
200 225.29 186.59 141.19 166.47 1.88 182.12 182.01 1.09 192.98 190.41 5.13
200 100 486.61 415.95 330.52 381.25 1.35 414.23 411.46 2.18 418.68 418.04 11.48
200 470.68 407.27 339.75 373.47 1.92 400.76 398.18 2.17 408.94 406.36 17.38
300 458.86 392.36 327.42 364.31 3.00 390.41 389.97 2.30 396.74 394.95 20.16
400 455.93 386.82 327.51 362.83 3.88 387.63 389.83 2.36 390.20 388.69 13.95
500 250 1195.38 1057.93 949.65 1007.69 3.25 1062.99 1065.15 8.05 1067.74 1067.13 58.52
500 1170.13 1038.79 927.82 991.64 6.95 1037.89 1036.55 6.67 1045.27 1040.02 69.19
750 1158.40 1036.04 921.19 990.71 10.56 1043.01 1043.29 8.53 1041.21 1039.92 72.33
1000 1153.98 1026.42 918.45 985.13 14.17 1029.30 1028.34 9.71 1043.27 1040.64 88.20
1000 500 2372.10 2155.47 1979.65 2090.65 10.81 2153.46 2153.81 13.66 2169.29 2167.92 151.93
1000 2338.73 2130.04 1965.58 2075.21 20.56 2133.01 2132.86 19.45 2146.65 2145.80 186.37
1500 2323.82 2124.92 1973.03 2063.91 30.24 2116.10 2112.71 26.33 2125.68 2123.82 306.66
2000 2315.50 2115.62 1991.24 2061.12 40.06 2129.96 2130.49 34.37 2139.15 2137.68 355.30
2000 1000 4725.17 4386.88 4168.28 4299.25 37.99 4396.60 4398.16 42.95 4413.11 4413.68 437.40
2000 4676.85 4358.33 4151.69 4279.57 65.00 4375.92 4379.46 69.17 4379.41 4377.45 820.25
3000 4653.55 4344.27 4142.43 4266.76 99.39 4343.23 4345.99 98.43 4370.00 4365.32 1135.90
4000 4637.74 4334.58 4109.75 4256.80 132.17 4336.27 4334.29 129.59 4361.70 4361.22 1797.67
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$ n $ $ m $ CR Algorithm in [13] Algorithm 2 Algorithm 4
max min ave time(s) $ f(\bar{x}) $ $ f(\tilde{x}) $ time(s) $ f(\bar{x}) $ $ f(\tilde{x}) $ time(s)
10 5 27.23 24.46 9.12 15.41 0.17 25.26 24.46 0.30 25.06 24.62 0.11
10 18.84 17.24 5.53 10.38 0.16 15.75 15.73 0.07 16.65 15.99 0.08
15 20.70 18.58 4.88 10.12 0.19 16.98 16.98 0.10 18.23 17.68 0.16
20 19.61 16.58 4.27 9.07 0.19 12.76 12.30 0.08 14.91 12.75 0.24
50 25 125.01 100.04 64.56 82.91 0.36 100.11 98.81 0.65 100.83 97.12 1.45
50 119.37 97.73 58.83 78.78 0.46 102.54 102.52 0.51 96.28 96.24 1.93
75 116.26 91.54 61.52 77.06 0.65 93.45 91.85 0.56 93.50 92.08 2.00
100 111.27 90.98 57.09 74.20 0.80 87.20 85.38 0.56 90.22 88.04 2.40
100 50 248.61 203.88 149.11 179.52 0.64 205.35 203.69 1.11 206.61 206.25 5.57
100 237.30 201.19 148.41 174.23 0.91 194.10 198.58 1.14 198.03 195.88 5.19
150 229.82 189.37 143.86 169.15 1.60 190.54 189.85 1.17 191.68 191.02 6.89
200 225.29 186.59 141.19 166.47 1.88 182.12 182.01 1.09 192.98 190.41 5.13
200 100 486.61 415.95 330.52 381.25 1.35 414.23 411.46 2.18 418.68 418.04 11.48
200 470.68 407.27 339.75 373.47 1.92 400.76 398.18 2.17 408.94 406.36 17.38
300 458.86 392.36 327.42 364.31 3.00 390.41 389.97 2.30 396.74 394.95 20.16
400 455.93 386.82 327.51 362.83 3.88 387.63 389.83 2.36 390.20 388.69 13.95
500 250 1195.38 1057.93 949.65 1007.69 3.25 1062.99 1065.15 8.05 1067.74 1067.13 58.52
500 1170.13 1038.79 927.82 991.64 6.95 1037.89 1036.55 6.67 1045.27 1040.02 69.19
750 1158.40 1036.04 921.19 990.71 10.56 1043.01 1043.29 8.53 1041.21 1039.92 72.33
1000 1153.98 1026.42 918.45 985.13 14.17 1029.30 1028.34 9.71 1043.27 1040.64 88.20
1000 500 2372.10 2155.47 1979.65 2090.65 10.81 2153.46 2153.81 13.66 2169.29 2167.92 151.93
1000 2338.73 2130.04 1965.58 2075.21 20.56 2133.01 2132.86 19.45 2146.65 2145.80 186.37
1500 2323.82 2124.92 1973.03 2063.91 30.24 2116.10 2112.71 26.33 2125.68 2123.82 306.66
2000 2315.50 2115.62 1991.24 2061.12 40.06 2129.96 2130.49 34.37 2139.15 2137.68 355.30
2000 1000 4725.17 4386.88 4168.28 4299.25 37.99 4396.60 4398.16 42.95 4413.11 4413.68 437.40
2000 4676.85 4358.33 4151.69 4279.57 65.00 4375.92 4379.46 69.17 4379.41 4377.45 820.25
3000 4653.55 4344.27 4142.43 4266.76 99.39 4343.23 4345.99 98.43 4370.00 4365.32 1135.90
4000 4637.74 4334.58 4109.75 4256.80 132.17 4336.27 4334.29 129.59 4361.70 4361.22 1797.67
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