Skip to main content

Table 2 Bounds for BoxQP relaxations

From: Tight compact extended relaxations for nonconvex quadratic programming problems with box constraints

Name (spar\(\ldots \)) \(z_{\mathcal {M}}\) \(z_{\mathcal {M}^2}\) \(z_{\mathcal {M}}^\circ \) \(z_{\mathcal {M}^2}^\circ \) \(z_{\mathcal {M}}^\bullet \) \(z_{\mathcal {M}^2}^\bullet \) \(z_{\text {BoxQP}}\)
020-100-1 \(-\)1066.00 \(-\)1038.38 \(-\)776.00 \(-\)706.89 \(-\)706.50 \(-\)706.50 \(-\)706.50
020-100-2 \(-\)1289.00 \(-\)1258.38 \(-\)951.50 \(-\)868.32 \(-\)880.25 \(-\)867.14 \(-\)856.50
020-100-3 \(-\)1168.50 \(-\)1142.00 \(-\)877.00 \(-\)772.13 \(-\)772.00 \(-\)772.00 \(-\)772.00
030-060-1 \(-\)1454.75 \(-\)1430.00 \(-\)761.50 \(-\)725.11 \(-\)730.06 \(-\)714.21 \(-\)706.00
030-060-2 \(-\)1699.50 \(-\)1668.25 \(-\)1449.00 \(-\)1379.18 \(-\)1385.50 \(-\)1379.18 \(-\)1377.17
030-060-3 \(-\)2047.00 \(-\)2006.50 \(-\)1388.00 \(-\)1315.19 \(-\)1323.56 \(-\)1305.97 \(-\)1293.50
030-070-1 \(-\)1569.00 \(-\)1547.25 \(-\)738.24 \(-\)704.17 \(-\)703.86 \(-\)688.50 \(-\)654.00
030-070-2 \(-\)1940.25 \(-\)1888.25 \(-\)1461.00 \(-\)1318.22 \(-\)1321.75 \(-\)1315.82 \(-\)1313.00
030-070-3 \(-\)2302.75 \(-\)2251.12 \(-\)1784.50 \(-\)1677.21 \(-\)1695.00 \(-\)1677.00 \(-\)1657.40
030-080-1 \(-\)2107.50 \(-\)2072.00 \(-\)1030.85 \(-\)987.81 \(-\)988.93 \(-\)967.73 \(-\)952.73
030-080-2 \(-\)2178.25 \(-\)2158.12 \(-\)1627.50 \(-\)1597.00 \(-\)1597.00 \(-\)1597.00 \(-\)1597.00
030-080-3 \(-\)2403.50 \(-\)2376.25 \(-\)1870.00 \(-\)1809.78 \(-\)1813.50 \(-\)1809.78 \(-\)1809.78
030-090-1 \(-\)2423.50 \(-\)2385.12 \(-\)1371.00 \(-\)1298.70 \(-\)1296.50 \(-\)1296.50 \(-\)1296.50
030-090-2 \(-\)2667.00 \(-\)2622.75 \(-\)1607.00 \(-\)1474.93 \(-\)1478.00 \(-\)1470.64 \(-\)1466.84
030-090-3 \(-\)2538.25 \(-\)2499.38 \(-\)1585.00 \(-\)1494.88 \(-\)1494.00 \(-\)1494.00 \(-\)1494.00
030-100-1 \(-\)2602.00 \(-\)2541.50 \(-\)1391.50 \(-\)1242.51 \(-\)1235.38 \(-\)1227.38 \(-\)1227.12
030-100-2 \(-\)2729.25 \(-\)2698.88 \(-\)1350.00 \(-\)1270.51 \(-\)1260.50 \(-\)1260.50 \(-\)1260.50
030-100-3 \(-\)2751.75 \(-\)2703.75 \(-\)1640.00 \(-\)1526.60 \(-\)1541.50 \(-\)1524.07 \(-\)1511.05
040-030-1 \(-\)1088.00 \(-\)1067.00 \(-\)853.00 \(-\)839.50 \(-\)839.50 \(-\)839.50 \(-\)839.50
040-030-2 \(-\)1635.00 \(-\)1617.75 \(-\)1461.00 \(-\)1429.36 \(-\)1431.50 \(-\)1429.36 \(-\)1429.00
040-030-3 \(-\)1303.25 \(-\)1297.12 \(-\)1110.50 \(-\)1086.00 \(-\)1086.00 \(-\)1086.00 \(-\)1086.00
040-040-1 \(-\)1606.25 \(-\)1575.50 \(-\)883.57 \(-\)857.50 \(-\)856.82 \(-\)847.93 \(-\)837.00
040-040-2 \(-\)1920.75 \(-\)1895.75 \(-\)1500.00 \(-\)1428.79 \(-\)1428.00 \(-\)1428.00 \(-\)1428.00
040-040-3 \(-\)2039.75 \(-\)2017.25 \(-\)1211.45 \(-\)1184.06 \(-\)1193.00 \(-\)1179.26 \(-\)1173.50
040-050-1 \(-\)2146.25 \(-\)2120.88 \(-\)1209.50 \(-\)1159.72 \(-\)1157.00 \(-\)1154.73 \(-\)1154.50
040-050-2 \(-\)2357.25 \(-\)2334.88 \(-\)1493.50 \(-\)1439.17 \(-\)1435.50 \(-\)1432.04 \(-\)1430.98
040-050-3 \(-\)2616.00 \(-\)2603.00 \(-\)1672.50 \(-\)1653.63 \(-\)1658.00 \(-\)1653.63 \(-\)1653.63
040-060-1 \(-\)2872.00 \(-\)2817.88 \(-\)1451.23 \(-\)1392.62 \(-\)1390.40 \(-\)1365.00 \(-\)1322.67
040-060-2 \(-\)2917.50 \(-\)2872.62 \(-\)2106.00 \(-\)2010.40 \(-\)2014.00 \(-\)2006.03 \(-\)2004.23
040-060-3 \(-\)3434.00 \(-\)3386.12 \(-\)2566.50 \(-\)2454.65 \(-\)2454.50 \(-\)2454.50 \(-\)2454.50
040-070-1 \(-\)3144.00 \(-\)3070.12 \(-\)1757.00 \(-\)1611.33 \(-\)1605.00 \(-\)1605.00
040-070-2 \(-\)3369.25 \(-\)3323.00 \(-\)1940.50 \(-\)1871.20 \(-\)1867.50 \(-\)1867.50
040-070-3 \(-\)3760.25 \(-\)3724.50 \(-\)2527.00 \(-\)2441.81 \(-\)2444.00 \(-\)2436.50
040-080-1 \(-\)3846.50 \(-\)3788.62 \(-\)2000.00 \(-\)1844.24 \(-\)1838.50 \(-\)1838.50
040-080-2 \(-\)3833.00 \(-\)3775.38 \(-\)2078.00 \(-\)1964.38 \(-\)1952.50 \(-\)1952.50
040-080-3 \(-\)4361.50 \(-\)4311.12 \(-\)2666.63 \(-\)2556.93 \(-\)2561.50 \(-\)2545.50
040-090-1 \(-\)4376.75 \(-\)4325.50 \(-\)2253.00 \(-\)2145.50 \(-\)2135.50 \(-\)2135.50
040-090-2 \(-\)4357.75 \(-\)4304.38 \(-\)2278.50 \(-\)2148.73 \(-\)2123.29 \(-\)2113.00
040-090-3 \(-\)4516.75 \(-\)4453.38 \(-\)2664.50 \(-\)2550.19 \(-\)2540.00 \(-\)2535.00
040-100-1 \(-\)5009.75 \(-\)4932.12 \(-\)2687.50 \(-\)2489.87 \(-\)2487.50 \(-\)2476.38
040-100-2 \(-\)4902.75 \(-\)4855.25 \(-\)2189.17 \(-\)2145.04 \(-\)2146.25 \(-\)2102.50
040-100-3 \(-\)5075.75 \(-\)5017.25 \(-\)2237.58 \(-\)2166.61 \(-\)2192.17 \(-\)1866.07
050-030-1 \(-\)1858.25 \(-\)1837.75 \(-\)1359.00 \(-\)1324.82 \(-\)1324.50 \(-\)1324.50
050-030-2 \(-\)2334.00 \(-\)2324.62 \(-\)1695.00 \(-\)1669.28 \(-\)1669.00 \(-\)1668.00
050-030-3 \(-\)2107.25 \(-\)2093.75 \(-\)1498.50 \(-\)1456.21 \(-\)1461.00 \(-\)1453.61
050-040-1 \(-\)2632.00 \(-\)2580.62 \(-\)1490.50 \(-\)1415.59 \(-\)1411.00 \(-\)1411.00
050-040-2 \(-\)2923.25 \(-\)2891.88 \(-\)1832.50 \(-\)1749.01 \(-\)1753.50 \(-\)1745.76
050-040-3 \(-\)3273.50 \(-\)3236.00 \(-\)2186.00 \(-\)2096.04 \(-\)2094.50 \(-\)2094.50
050-050-1 \(-\)3536.00 \(-\)3506.25 \(-\)1450.99 \(-\)1415.71 \(-\)1409.72 \(-\)1198.41
050-050-2 \(-\)3500.50 \(-\)3467.12 \(-\)1855.04 \(-\)1806.44 \(-\)1776.81 \(-\)1776.00
050-050-3 \(-\)4119.75 \(-\)4052.12 \(-\)2293.88 \(-\)2151.14 \(-\)2138.34 \(-\)2106.10
060-020-1 \(-\)1757.25 \(-\)1745.50 \(-\)1223.50 \(-\)1212.00 \(-\)1212.00 \(-\)1212.00
060-020-2 \(-\)2238.25 \(-\)2230.00 \(-\)1925.50 \(-\)1925.50 \(-\)1925.50 \(-\)1925.50
060-020-3 \(-\)2098.75 \(-\)2081.00 \(-\)1518.00 \(-\)1483.42 \(-\)1483.00 \(-\)1483.00
070-025-1 \(-\)3832.75 \(-\)3788.88 \(-\)2615.50 \(-\)2553.18 \(-\)2545.00 \(-\)2538.91
070-025-2 \(-\)3248.00 \(-\)3232.88 \(-\)1935.00 \(-\)1895.01 \(-\)1888.50 \(-\)1888.00
070-025-3 \(-\)4167.25 \(-\)4148.38 \(-\)2880.00 \(-\)2819.92 \(-\)2819.25 \(-\)2812.28
070-050-1 \(-\)7210.75 \(-\)7151.12 \(-\)3462.90 \(-\)3411.63 \(-\)3356.00 \(-\)3252.50
070-050-2 \(-\)6620.00 \(-\)6573.88 \(-\)3409.56 \(-\)3341.97 \(-\)3296.00 \(-\)3296.00
070-050-3 \(-\)7522.00 \(-\)7473.88 \(-\)4419.50 \(-\)4308.40 \(-\)4306.50 \(-\)4306.50
070-075-1 \(-\)11,647.75 \(-\)11,578.12 \(-\)5089.09 \(-\)5010.60 \(-\)5003.67 \(-\)4655.50
070-075-2 \(-\)10,884.75 \(-\)10,793.38 \(-\)4583.49 \(-\)4469.11 \(-\)4504.92 \(-\)3865.15
070-075-3 \(-\)11,262.25 \(-\)11,162.38 \(-\)4944.31 \(-\)4826.89 \(-\)4862.75 \(-\)4329.40
080-025-1 \(-\)4840.75 \(-\)4829.12 \(-\)3195.00 \(-\)3157.23 \(-\)3157.00 \(-\)3157.00
080-025-2 \(-\)4378.50 \(-\)4351.00 \(-\)2432.78 \(-\)2383.57 \(-\)2361.62 \(-\)2312.34
080-025-3 \(-\)5130.25 \(-\)5102.88 \(-\)3156.00 \(-\)3108.04 \(-\)3101.00 \(-\)3090.88
080-050-1 \(-\)9783.25 \(-\)9696.62 \(-\)4154.85 \(-\)4042.38 \(-\)4025.80 \(-\)3448.10
080-050-2 \(-\)9270.00 \(-\)9205.50 \(-\)4595.53 \(-\)4498.70 \(-\)4450.50 \(-\)4449.20
080-050-3 \(-\)10,029.75 \(-\)9967.25 \(-\)5216.82 \(-\)5044.25 \(-\)4961.27 \(-\)4886.00
080-075-1 \(-\)15,250.75 \(-\)15,154.75 \(-\)6683.91 \(-\)6579.91 \(-\)6601.92 \(-\)5896.00
080-075-2 \(-\)14,246.50 \(-\)14,146.62 \(-\)6058.35 \(-\)5945.51 \(-\)5953.17 \(-\)5341.00
080-075-3 \(-\)14,961.50 \(-\)14,860.88 \(-\)6776.83 \(-\)6652.30 \(-\)6584.00 \(-\)5980.50
090-025-1 \(-\)6171.50 \(-\)6135.25 \(-\)3526.12 \(-\)3469.80 \(-\)3423.78 \(-\)3372.50
090-025-2 \(-\)6015.00 \(-\)5978.38 \(-\)3668.68 \(-\)3588.62 \(-\)3550.65 \(-\)3500.29
090-025-3 \(-\)6698.25 \(-\)6681.88 \(-\)4367.98 \(-\)4341.99 \(-\)4299.00 \(-\)4299.00
090-050-1 \(-\)12,584.00 \(-\)12,522.38 \(-\)5611.51 \(-\)5547.69 \(-\)5468.90 \(-\)5152.00
090-050-2 \(-\)11,920.50 \(-\)11,,851.38 \(-\)5573.10 \(-\)5514.60 \(-\)5404.36 \(-\)5386.50
090-050-3 \(-\)12,514.00 \(-\)12452.50 \(-\)6439.11 \(-\)6345.88 \(-\)6230.59 \(-\)6151.00
090-075-1 \(-\)19054.25 \(-\)18,944.50 \(-\)8016.93 \(-\)7899.63 \(-\)7944.92 \(-\)6267.45
090-075-2 \(-\)18,245.50 \(-\)18,132.50 \(-\)7406.32 \(-\)7290.15 \(-\)7334.75 \(-\)5647.50
090-075-3 \(-\)18,929.50 \(-\)18,823.50 \(-\)7974.10 \(-\)7866.58 \(-\)7908.50 \(-\)6450.00
100-025-1 \(-\)7660.75 \(-\)7611.38 \(-\)4255.79 \(-\)4177.38 \(-\)4116.48 \(-\)4027.50
100-025-2 \(-\)7338.50 \(-\)7303.12 \(-\)4002.76 \(-\)3975.38 \(-\)3906.07 \(-\)3892.56
100-025-3 \(-\)7942.25 \(-\)7894.75 \(-\)4581.40 \(-\)4504.17 \(-\)4459.25 \(-\)4453.50
100-050-1 \(-\)15,415.75 \(-\)15,341.75 \(-\)6474.28 \(-\)6407.46 \(-\)6366.84 \(-\)5490.00
100-050-2 \(-\)14,920.50 \(-\)14,814.62 \(-\)6665.38 \(-\)6573.51 \(-\)6504.93 \(-\)5866.00
100-050-3 \(-\)15,564.25 \(-\)15,480.12 \(-\)7179.24 \(-\)7080.22 \(-\)7031.72 \(-\)6485.00
100-075-1 \(-\)23,387.50 \(-\)23,277.12 \(-\)9645.89 \(-\)9504.28 \(-\)9551.75 \(-\)7384.20
100-075-2 \(-\)22,440.00 \(-\)22,307.00 \(-\)8927.68 \(-\)8768.39 \(-\)8826.42 \(-\)6755.50
100-075-3 \(-\)23,243.50 \(-\)23,109.62 \(-\)9692.79 \(-\)9553.26 \(-\)9614.25 \(-\)7554.00
125-025-1 \(-\)12,251.00 \(-\)12,184.75 \(-\)6244.26 \(-\)6164.92 \(-\)6118.03 \(-\)5572.00
125-025-2 \(-\)12,732.00 \(-\)12,662.62 \(-\)6596.36 \(-\)6462.89 \(-\)6401.29 \(-\)6156.06
125-025-3 \(-\)12,650.75 \(-\)12,627.50 \(-\)7055.47 \(-\)7019.67 \(-\)6923.00 \(-\)6815.50
125-050-1 \(-\)24,993.00 \(-\)24,880.25 \(-\)11,030.48 \(-\)10,879.26 \(-\)10,879.42 \(-\)9308.38
125-050-2 \(-\)24,810.50 \(-\)24,669.38 \(-\)10,388.46 \(-\)10,215.57 \(-\)10,273.75 \(-\)8395.00
125-050-3 \(-\)24,424.50 \(-\)24,308.00 \(-\)10,126.01 \(-\)9999.21 \(-\)10,032.50 \(-\)8343.91
125-075-1 \(-\)38,202.00 \(-\)38,058.12 \(-\)16,181.85 \(-\)16,009.38 \(-\)16,053.67 \(-\)12,330.00
125-075-2 \(-\)37,466.75 \(-\)37,341.38 \(-\)15,158.92 \(-\)15,025.72 \(-\)15,088.58 \(-\)10,382.47
125-075-3 \(-\)36,202.25 \(-\)36,033.00 \(-\)14,002.17 \(-\)13,857.12 \(-\)13,917.67 \(-\)9635.50
  1. All values in columns \(z_{\mathcal {M}}\), \(z_{\mathcal {M}^2}\), \(z_{\mathcal {M}}^\circ \), \(z_{\mathcal {M}^2}^\circ \), and \(z_{\text {BoxQP}}\) were taken from the computational study of Bonami et al. [3]