Article Contents
Article Contents

• * Corresponding author: Mohammad Keyanpour
• 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:

• 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

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 —

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

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 —
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