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On fractional quadratic optimization problem with two quadratic constraints

  • * Corresponding author: Mohammad Keyanpour

    * Corresponding author: Mohammad Keyanpour 
Abstract Full Text(HTML) Figure(1) / Table(4) Related Papers Cited by
  • In this paper, we study the problem of minimizing the ratio of two quadratic functions subject to two quadratic constraints in the complex space. Using the classical Dinkelbach method, we transform the problem into a parametric nonlinear equation. We show that an optimal parameter can be found by employing the S-procedure and semidefinite relaxation technique. A key element to solve the original problem is to use the rank-one decomposition procedure. Finally, within the new algorithm, semidefinite relaxation is compared with the bisection method for finding the root on several examples. For further comparison, the solution of fmincon command of MATLAB also is reported.

    Mathematics Subject Classification: 90C32, 90C20, 90C26, 90C22.

    Citation:

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  • Figure 1.  Multi-relays cooperative cognitive communication model [7]

    Table 1.  Numerical Results for Example 1

    SDO Method Bisection Method fmincon
    n density λ* fvalue time(s) λ* fvalue time(s) fvalue time(s)
    50 1 1.201032e-01 1.201036e-01 0.2703 1.201032e-01 1.201036e-01 5.0101 1.397555e-01 0.0117
    100 1 1.881241e-01 1.881249e-01 1.086:3 1.881241e-01 1.881249e-01 20.5448 1.987974e-01 0.0139
    150 1 1.643814e-01 1.643875e-01 2.5524 1.643814e-01 1.643875e-01 50.1663 1.698622e-01 0.0174
    200 1 1.558427e-01 1.558441e-01 5.9194 1.558427e-01 1.558441e-01 153.8498 1.755857e-01 0.0194
    250 1 2.366953e-01 2.366986e-01 8.1471 2.366953e-01 2.366986e-01 226.1209 2.841313e-01 0.0269
    50 0.5 1.193231e-02 1.193236e-02 0.2473 1.193231e-02 1.193236e-02 4.7561 1.958082e-02 0.0109
    100 0.5 1.446475e-01 1.446483e-01 0.9903 1.446475e-01 1.446483e-01 18.3720 1.596917e-01 0.0128
    150 0.5 1.024333e-01 1.024354e-01 2.4031 1.024333e-01 1.024354e-01 43.7765 1.129475e-01 0.0167
    200 0.5 2.134589e-01 2.134593e-01 5.3849 2.371164e-01 0.0173
    250 0.5 2.103671e-01 2.103684e-01 7.9862 2.278996e-01 0.0238
    300 0.5 1.507763e-01 1.507789e-01 14.7993 1.631893e-01 0.0237
    50 0.25 1.097243e-01 1.097248e-01 0.2444 1.097243e-01 1.097248e-01 4.4489 2.862603e-01 0.0098
    100 0.25 3.690304e-01 3.690309e-01 0.9323 3.690304e-01 3.690309e-01 12.7842 3.791923e-01 0.0121
    150 0.25 1.816562e-01 1.816565e-01 2.7160 1.816562^01 1.816565e-01 64.4395 1.944783e-01 0.0161
    200 0.25 2.123675e-01 2.123678e-01 5.0127 2.897431e-01 0.0159
    250 0.25 2.325119e-01 2.325124e-01 7.5622 2.666913e-01 0.0198
    300 0.25 1.394089e-01 1.394093e-01 14.3163 1.534687e-01 0.0219
    50 0.1 2.029650e-01 2.029654e-01 0.2021 2.029650e-01 2.029654e-01 3.8651 2.273829e-01 0.0086
    100 0.1 4.695384e-01 4.695387e-01 0.8796 4.695384e-01 4.695387e-01 12.1470 4.812845e-01 0.0117
    150 0.1 1.325870e-01 1.325874e-01 2.5595 1.325870e-01 1.325874e-01 49.9254 1.470531e-01 0.0144
    200 0.1 1.442915e-01 1.442918e-01 4.8677 1.700139e-01 0.0151
    250 0.1 4.210943e-01 4.210946e-01 7.1247 4.512736e-01 0.0179
    300 0.1 1.547663e-01 1.547C71e-01 13.9812 1.673333e-01 0.0204
    350 0.1 3.037856e-01 3.037861e-01 19.2776 3.293677e-01 0.0245
    100 0.01 3.132863e-02 3.132868e-02 0.6293 3.132863e-02 3.132868e-02 11.4868 3.693282e-02 0.0117
    150 0.01 1.423572e-02 1.423577e-02 1.5306 1.423572e-02 1.423577e-02 37.9483 3.092266e-02 0.0128
    200 0.01 3.145664e-01 3.145669e-01 4.2319 3.364780e-01 0.0139
    250 0.01 2.597103e-02 2.597105e-02 6.8963 2.777693e-02 0.0167
    300 0.01 1.201043e-01 1.201049e-01 13.4639 1.537038e-01 0.0192
    350 0.01 2.167836e-02 2.167842e-02 18.6749 2.478134e-02 0.0235
    400 0.01 4.531146^01 4.531156e-01 24.1313 4.972134e-01 0.0263
     | Show Table
    DownLoad: CSV

    Table 2.  Numerical Results for Example 1

    SDO Method Bisection Method fmincon
    n density λ* fvalue time(s) λ* fvalue time(s) fvalue time(s)
    50 1 9.963074e-02 9.963075e-02 0.2313 9.963074-02 9.963075e-02 3.8289
    100 1 2.954031e-02 2.954037e-02 1.0432 2.954031e-02 2.954037e-02 18.6609
    150 1 1.079322e-01 L079326e-01 2.7307 1.079322e-01 1.079326e-01 47.8104
    200 1 1.594683e-01 1.594686e-01 5.8439
    250 1 2.374876e-01 2.374887e-01 8.5416
    50 0.5 2.644901e-02 2.644909e-02 0.2574 2.644901e-02 2.644909e-02 4.2066
    100 0.5 2.299581e-01 2.299585e-01 1.0217 2.299581e-01 2.299585e-01 19.3125
    150 0.5 1.444501e-01 1.444506e-01 2.2918 1.444501e-01 1.444506e-01 44.9254
    200 0.5 3.724168e-01 3.724173e-01 5.4688
    250 0.5 2.552464e-01 2.552473e-01 8.2069
    300 0.5 2.941125e-01 2.941131e-01 16.0230
    50 0.25 1.076935e-02 1.076939e-02 0.2603 1.076935e-02 1.076939e-02 5.0748
    100 0.25 1.768924e-02 1.768927e-02 0.9174 1.768924e-02 1.768927e-02 12.5113
    150 0.25 2.993642e-01 2.993644e-01 2.0326 2.993642e-01 2.993644e-01 41.9702
    200 0.25 3.366870e-01 3.366874e-01 4.9010
    250 0.25 1.883748e-01 1.883755e-02 7.2461
    300 0.25 1.649437e-01 1.649446e-01 15.3795
    50 0.1 1.692921e-01 1.692924e-01 0.2097 1.692921e-01 1.692924e-01 3.9190
    100 0.1 1.942553e-01 1.942554e-01 0.9273 1.942553e-01 1.942554e-01 11.9908
    150 0.1 2.121414e-02 2.121419e-02 1.9469 2.121414e-02 2.121419e-02 40.4315
    200 0.1 2.737760e-01 2.737764e-01 4.8436
    250 0.1 1.803547e-01 1803554e-01 7.3638
    300 0.1 3.003607e-01 3.003612e-01 14.2789
    350 0.1 2.941177e-01 2.941187e-01 20.2413
    100 0.01 1.462551e-02 1.462558e-02 0.6089 1.462558e-02 1.462558e-02 11.6858
    150 0.01 1.371044e-01 1.371049e-01 1.4294 1.371044e-01 1.371049e-01 38.2468
    200 0.01 2.910353e-01 2.910355e-01 4.4201
    250 0.01 1.110103e-01 1.110109e-01 6.5731
    300 0.01 3.831475e-01 3.831484e-01 13.8774
    350 0.01 1.301978e-01 1.301985e-01 19.2763
    400 0.01 2.705236e-01 2.705243e-01 23.9422
     | Show Table
    DownLoad: CSV

    Table 3.  Numerical Results for Example 2

    SDO Method Bisection Method fmincon
    n density λ* fvalue time(s) λ* fvalue time(s) fvalue time(s)
    50 1 1.073815e-01 1.073819e-01 0.2603 1.073815e-01 1.073819e-01 5.6108 1.331134e-01 0.0097
    100 1 8.052910e-02 8.052916e-02 1.0407 8.052910e-02 8.052916e-02 23.5367 8.699541e-02 0.0117
    150 1 3.159523e-01 3.159529e-01 2.6979 3.159523e-01 3.159529e-01 57.1506 3.692334e-02 0.0131
    200 1 9.654340e-01 9.654352e-01 5.7237 9.654340e-01 9.654352e-01 120.0627 9.845699e-01 0.U192
    250 1 5.314459e-01 5.314467e-01 7.9846 5.314459e-01 5.314467e-01 146.7938 5.983331e-01 0.0247
    50 0.5 6.030734e-02 6.030738e-02 0.2465 6.030734e-02 6.030738e-02 5.2755 6.404861e-02 0.0114
    100 0.5 7.375045e-02 7.375047e-02 0.9825 7.375045e-02 7.375047e-02 21.3846 7.915205e-02 0.0102
    150 0.5 2.523155e-01 2.523164e-01 2.8862 2.523155e-01 2.523164e-01 50.9374 2.689658e-01 0.0133
    2U0 0.5 1.997833e-01 1.997841e-01 5.3377 2.321445e-01 0.0158
    250 0.5 4.256789e-01 4.256796e-01 7.7639 4.433670e-01 0.0239
    300 0.5 6.730019e-01 6.730022e-01 12.2279 6.846389e-01 0.0268
    50 0.25 1.119725e-01 1.119727e-01 0.2428 1.119725e-01 1.119727e-01 5.0119 1.289442e-01 0.0090
    100 0.25 4.884773e-02 4.884778e-02 0.9657 4.884773e-02 4.884778e-02 14.003-1 5.102978e-01 0.0109
    150 0.25 2.374415e-01 2.374418e-01 2.7611 2.374415e-01 2.374418e-01 47.1437 2.531947e-01 0.0124
    200 0.25 3.264711e-02 3.264719e-02 5.1312 3.803010e-02 0.0144
    250 0.25 5.412756e-01 5.412759e-01 7.3988 5.732258e-01 0.0216
    300 0.25 1.222439e-01 1.222445e-01 12.0333 1.986063e-01 0.0253
    50 0.1 1.833330e-01 1.833334e-01 0.2236 1.833330e-01 1.833334e-01 4.9949 1.970122e-01 0.0094
    100 0.1 3.380843e-01 3.380847e-01 0.8375 3.380843e-01 3.380847e-01 17.1359 3.426797e-01 0.0112
    150 0.1 3.348966e-01 3.348969e-01 2.4617 3.348966e-01 3.348969e-01 46.9201 3.532487e-01 0.0135
    200 0.1 1.355464e-01 1.355466e-01 4.8699 1.836288e-01 0.0137
    250 0.1 2.002679e-01 2.002685e-01 7.1009 2.321444e-01 0.0205
    300 0.1 3.191755e-01 3.191761e-01 11.8233 3.274615e-01 0.0242
    350 0.1 8.630814e-02 8.630823e-02 18.7753 8.949326e-01 0.0276
    100 0.01 2.322450e-02 2.322456e-02 0.7865 2.322450e-02 2.322456e-02 16.7437 2.481309e-02 0.0104
    150 0.01 5.444251e-02 5.444253e-02 2.1999 5.444251e-02 5.444253e-02 43.2585 5.613323e-02 0.0126
    200 0.01 2.510731e-01 2.510735e-01 4.5176 2.713509e-01 0.0128
    250 0.01 3.722411e-01 3.722423e-01 6.9452 3.830029e-01 0.0132
    300 0.01 1.676625e-01 1.676632e-01 11.3290 1.821748e-01 0.0229
    350 0.01 1.853554e-01 1.853560e-01 19.5423 1.962172e-01 0.0238
    400 0.01 4.531146e-01 4.531156e-01 24.1313 4.972134e-01 0.0263
     | Show Table
    DownLoad: CSV

    Table 4.  Numerical Results for Example 2

    SDO Method Bisection Method fmincon
    n density λ* fvalue time(s) λ* fvalue time(s) fvalue time(s)
    50 1 1.579354e-01 1.579359e-01 0.2742 1.579354e-01 1.579359e-01 5.1885
    100 1 5.214084e-02 5.214087e-02 0.9871 5.214084e-02 5.214087e-02 23.9418
    150 1 2.951871e-02 2.951876e-02 2.5428 2.951871e-02 2.951876e-02 56.8249
    200 1 4.437936e-01 4.437943e-01 5.2411
    250 1 1.302289e-01 1.302295e-01 7.5028
    50 0.5 2.984366e-02 2.984369e-02 0.2527 2.984366e-02 2.984369e-02 5.0748
    100 0.5 2.547812e-02 2.547816e-02 0.7961 2.547812e-02 2.547816e-02 19.8679
    150 0.5 1.330274e-01 1.330278e-01 2.2318 1.330274e-01 1.330278e-01 55.3997
    200 0.5 4.219078e-01 4.219083e-01 5.2065
    250 0.5 2.903107e-01 2.903111e-01 7.1589
    300 0.5 3.600279e-02 3.600286e-01 13.2012
    50 0.25 1.742665e-02 1.742669e-02 0.2375 1.742665e-02 1.742669e-02 4.9128
    100 0.25 6.999637e-02 6.999639e-02 0.7539 6.999637e-02 6.999639e-02 19.2247
    150 0.25 2.181804e-02 2.181807e-02 2.1246 2.181804e-02 2.181807e-02 54.8697
    200 0.25 1.898546e-01 1.898552e-01 4.9963 - -
    250 0.25 4.199379e-01 4.199380e-01 6.8883
    300 0.25 2.342168e-01 2.342173e-01 12.9771
    50 0.1 2.254114e-02 2.254117e-02 0.2162 2.254114e-02 2.254117e-02 4.3246
    100 0.1 1.947362e-02 1.947365e-02 0.6987 1.947362e-02 1.947365e-02 18.9781
    150 0.1 1.596201e-01 1.596206e-01 1.9342 1.596201e-01 1.596206e-01 52.7639
    200 0.1 2.603211e-01 2.603213e-01 4.5136
    250 0.1 2.274938e-01 2.274946e-01 6.2734
    300 0.1 1.757963e-01 1.757970e-01 12.4938
    100 0.01 1.977456e-02 1.977458e-02 0.7276 1.977456e-02 1.977458e-02 4.1807
    150 0.01 6.123555e-02 6.123557e-02 1.5383 6.123555e-02 6.123557e-02 50.9088
    200 0.01 9.402653e-02 9.402659e-02 4.2944
    250 0.01 2.351025e-01 2.351028e-01 5.8472
    300 0.01 1.639947e-01 1.639959e-01 11.8967
    350 0.01 2.486289e-01 2.486296e-01 20.1709
    400 0.01 2.519726e-01 2.519735e-01 25.0546
     | Show Table
    DownLoad: CSV
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