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Tighter quadratically constrained convex reformulations for semi-continuous quadratic programming

  • * Corresponding author: Zhongyi Jiang

    * Corresponding author: Zhongyi Jiang

This research was supported by the National Natural Science Foundation of China under Grants 11671300

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  • The paper proposes a novel class of quadratically constrained convex reformulations (QCCR) for semi-continuous quadratic programming. We first propose the class of QCCR for the studied problem. Next, we discuss how to polynomially find the best reformulation corresponding with the tightest continuous bound within this class. The properties of the proposed QCCR are then studied. Finally, preliminary computational experiments are conducted to illustrate the effectiveness of the proposed approach.

    Mathematics Subject Classification: Primary: 90C11, 90C20; Secondary: 90C22.

    Citation:

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  • Table 1.  Comparison results of perspective reformulation and QCCR for Problem $ \rm{(TP)} $

    $ n $ $ K $ $ {\rm T_{PR}} $ $ {\rm T_{QCP}} $ $ ({\rm PR_{SOCP}}) $ $ ({\rm QCP}) $
    gap(%) time nodes gap(%) time nodes
    $ 200^+ $ 6 18.94 141.81 3.02 1800.00 8305 0.02 727.71 533846
    $ 200^+ $ 8 18.85 138.23 3.15 1800.00 7726 0.91 1753.91 1777822
    $ 200^+ $ 10 18.74 110.77 3.49 1800.00 6052 1.67 1800.00 2084181
    $ 200^+ $ 12 18.35 124.77 3.52 1800.01 6100 1.86 1800.00 1941049
    $ 200^0 $ 6 20.40 124.90 34.75 1800.01 11038 27.09 1800.01 2183238
    $ 200^0 $ 8 17.60 126.74 33.74 1800.01 10520 28.67 1800.01 2020237
    $ 200^0 $ 10 16.68 116.01 34.17 1800.00 6104 27.61 1800.00 2612416
    $ 200^0 $ 12 16.31 126.23 33.17 1800.00 8758 28.65 1800.01 2293060
    $ 200^- $ 6 18.78 129.74 58.70 1800.01 12628 50.00 1800.01 1990354
    $ 200^- $ 8 19.52 125.89 58.91 1800.00 11850 53.12 1800.01 2428332
    $ 200^- $ 10 19.00 125.75 58.75 1800.01 10990 55.25 1800.01 1911470
    $ 200^- $ 12 17.80 128.07 58.80 1800.00 11449 55.41 1800.01 1456779
    $ 300^+ $ 6 48.31 314.17 3.01 1447.96 4793 1.97 1800.01 1357610
    $ 300^+ $ 8 49.16 298.87 3.37 1445.44 3134 1.88 1441.80 1249562
    $ 300^+ $ 10 48.54 320.01 3.37 1454.77 2186 2.04 1800.01 1024344
    $ 300^+ $ 12 48.83 340.17 3.32 1502.79 2077 2.37 1800.01 811791
    $ 300^0 $ 6 46.55 298.79 40.90 1800.00 3321 32.88 1800.01 2435679
    $ 300^0 $ 8 43.08 295.82 40.66 1800.01 3452 34.04 1800.00 2132799
    $ 300^0 $ 10 39.67 301.40 40.49 1800.00 3146 34.83 1800.00 1855355
    $ 300^0 $ 12 42.47 279.30 40.24 1800.00 3648 35.28 1800.00 1701796
    $ 300^- $ 6 51.64 327.34 61.71 1800.00 4049 51.76 1800.01 1820914
    $ 300^- $ 8 51.08 294.60 61.29 1800.00 3859 53.30 1800.00 1806775
    $ 300^- $ 10 49.27 299.97 60.96 1800.00 3157 53.70 1800.00 1491932
    $ 300^- $ 12 50.44 276.10 60.52 1800.00 3824 55.42 1800.01 1238498
    $ 400^+ $ 6 106.29 655.85 4.53 1800.00 1955 2.91 1800.00 760852
    $ 400^+ $ 8 104.59 606.55 4.62 1800.00 1588 3.99 1800.01 546502
    $ 400^+ $ 10 111.16 558.01 4.43 1800.01 1602 3.16 1800.00 608266
    $ 400^+ $ 12 104.56 604.67 4.47 1800.00 1888 3.18 1800.00 687127
    $ 400^0 $ 6 110.19 603.64 35.42 1800.00 1919 31.80 1800.01 736142
    $ 400^0 $ 8 105.99 524.36 35.37 1800.00 1973 31.16 1800.00 800639
    $ 400^0 $ 10 112.22 498.27 35.33 1800.00 2127 32.78 1800.01 764762
    $ 400^0 $ 12 104.45 531.99 35.07 1800.01 2825 31.24 1800.00 954640
    $ 400^- $ 6 112.19 650.55 65.45 1800.00 1795 60.67 1800.01 571066
    $ 400^- $ 8 117.85 574.13 65.25 1800.01 1767 60.43 1800.00 695491
    $ 400^- $ 10 115.66 634.48 65.00 1800.00 2169 60.07 1800.00 707309
    $ 400^- $ 12 121.61 560.47 64.68 1800.00 2815 60.63 1800.00 831108
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