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Article Contents

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

• * Corresponding author: Zi Xu

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.

Mathematics Subject Classification: Primary: 90C06, 90C26; Secondary: 93E10.

 Citation:

• 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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