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September
2020, 10(3): 301-315.
doi: 10.3934/naco.2020003
On fractional quadratic optimization problem with two quadratic constraints
| 1. | Faculty of Mathematics Statistics and Computer Science, Semnan University, Semnan, Iran |
| 2. | Faculty of Mathematical Sciences and, Center of Excellence for Mathematical Modeling, Optimization and Combinatorial Computing (MMOCC), University of Guilan, Rasht, Iran |
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.
Keywords:
Fractional quadratic optimization,
Nonconvex problem,
S-procedure,
Semidefinite optimization,
Semidefinite relaxation.
Mathematics Subject Classification:
90C32, 90C20, 90C26, 90C22.
Citation:
Arezu Zare, Mohammad Keyanpour, Maziar Salahi. On fractional quadratic optimization problem with two quadratic constraints. Numerical Algebra, Control and Optimization,
2020, 10
(3)
: 301-315.
doi: 10.3934/naco.2020003
![]()
References:
| [1] |
R. A. Abrams,
Nonlinear programming in complex space: sufficient conditions and duality, Journal of Mathematical Analysis and Applications, 38 (1972), 619-632.
doi: 10.1016/0022-247X(72)90073-X. |
| [2] |
R. A. Abrams and A. Ben-Israel,
Nonlinear programming in complex space: necessary conditions, SIAM Journal on Control, 9 (1971), 606-620.
|
| [3] |
A. Aubry, V. Carotenuto and A. De Maio,
New results on generalized fractional programming problems with toeplitz quadratics, IEEE Signal Processing Letters, 23 (2016), 848-852.
|
| [4] |
A. Aubry, A. De Maio, Y. Huang and M. Piezzo,
Robust design of radar doppler filters, IEEE Transactions on Signal Processing, 64 (2016), 5848-5860.
doi: 10.1109/TSP.2016.2576423. |
| [5] |
T. Baleshan, A. Jayalath and J. Coetzee,
On power minimisation and snr maximisation of distributed beamforming in cooperative communication systems, Electronics Letters, 50 (2014), 712-713.
|
| [6] |
O. Besson,
Adaptive detection with bounded steering vectors mismatch angle, IEEE Transactions on Signal Processing, 55 (2007), 1560-1564.
doi: 10.1109/TSP.2006.890820. |
| [7] |
H. Cai, Y. Wang and T. Yi,
An approach for minimizing a quadratically constrained fractional quadratic problem with application to the communications over wireless channels, Optimization Methods and Software, 29 (2014), 310-320.
|
| [8] |
H. Chen, S. Shahbazpanahi and A. B. Gershman,
Filter-and-forward distributed beamforming for two-way relay networks with frequency selective channels, IEEE Transactions on Signal Processing, 60 (2011), 1927-1941.
doi: 10.1109/TSP.2011.2178842. |
| [9] |
J. Chen, H. Lai and S. Schaible,
Complex fractional programming and the charnes-cooper transformation, Journal of Optimization Theory and Applications, 126 (2005), 203-213.
doi: 10.1007/s10957-005-2669-y. |
| [10] |
A. De Maio, Y. Huang, D. P. Palomar, S. Zhang and A. Farina,
Fractional qcqp with applications in ml steering direction estimation for radar detection, IEEE Transactions on Signal Processing, 59 (2010), 172-185.
doi: 10.1109/TSP.2010.2087327. |
| [11] |
W. Dinkelbach,
On nonlinear fractional programming, Management Science, 13 (1967), 492-498.
doi: 10.1287/mnsc.13.7.492. |
| [12] |
S. Fallahi and M. Salahi,
On the complex fractional quadratic optimization with a quadratic constraint, Opsearch, 54 (2017), 94-106.
doi: 10.1007/s12597-016-0263-8. |
| [13] |
M. Grant and S. Boyd, CVX: Matlab Software for Disciplined Convex Programming, Version 2.1, 2014.
doi: 10.1145/2700586. |
| [14] |
J. Hsiao,
On the optimization of mti clutter rejection, IEEE Transactions on Aerospace and Electronic Systems, AES-10 (1974), 622-629.
|
| [15] |
Y. Huang and S. Zhang,
Complex matrix decomposition and quadratic programming, Mathematics of Operations Research, 32 (2007), 758-768.
doi: 10.1287/moor.1070.0268. |
| [16] |
R. Jagannathan,
On some properties of programming problems in parametric form pertaining to fractional programming, Management Science, 12 (1966), 609-615.
doi: 10.1287/mnsc.12.7.609. |
| [17] |
V. Jeyakumar, N. Q. Huy and G. Li, Necessary and sufficient conditions for s-lemma and nonconvex quadratic optimization, Optimization and Engineering, 10 (2009), 491.
doi: 10.1007/s11081-008-9076-9. |
| [18] |
U. T. Jönsson,
A lecture on the s-procedure, Lecture Note at the Royal Institute of Technology, Sweden, 23 (2001), 34-36.
|
| [19] |
N. Levinson,
Linear programming in complex space, Journal of Mathematical Analysis and Applications, 14 (1966), 44-62.
doi: 10.1016/0022-247X(66)90061-8. |
| [20] |
V.-B. Nguyen, R.-L. Sheu and Y. Xia,
An SDP approach for quadratic fractional problems with a two-sided quadratic constraint, Optimization Methods and Software, 31 (2016), 701-719.
doi: 10.1080/10556788.2015.1029575. |
| [21] |
I. Pólik and T. Terlaky,
A survey of the s-lemma, SIAM Review, 49 (2007), 371-418.
doi: 10.1137/S003614450444614X. |
| [22] |
M. Salahi and A. Zare,
SDO relaxation approach to fractional quadratic minimization with one quadratic constraint, Journal of Mathematical Modeling, 3 (2015), 1-13.
|
| [23] |
R. L. Sheu, An SDP approach for some quadratic fractional problems (nonlinear analysis and convex analysis). |
| [24] |
J. F. Sturm and S. Zhang,
On cones of nonnegative quadratic functions, Mathematics of Operations Research, 28 (2003), 246-267.
doi: 10.1287/moor.28.2.246.14485. |
| [25] |
K. Swarup and J. Sharma, Programming with linear fractional functionals in complex spaces, Cahiers du centre Études et de Recherche Opérationelle, 12 (1970), 103–109. |
| [26] |
T. Yi, L. Guo, K. Niu, H. Cai, J. Lin and W. Ai, Cooperative beamforming in cognitive radio network with hybrid relay, in 2012 19th International Conference on Telecommunications (ICT), IEEE, (2012), 1–5. |
| [27] |
A. Zhang and S. Hayashi,
Celis-Dennis-Tapia based approach to quadratic fractional programming problems with two quadratic constraints, Numer. Algebra, Control and Optim., 1 (2011), 83-98.
doi: 10.3934/naco.2011.1.83. |
| [28] |
X. Zhang, Z. He, X. Zhang and W. Peng,
High-performance beampattern synthesis via linear fractional semidefinite relaxation and quasi-convex optimization, IEEE Transactions on Antennas and Propagation, 66 (2018), 3421-3431.
|
show all references
References:
| [1] |
R. A. Abrams,
Nonlinear programming in complex space: sufficient conditions and duality, Journal of Mathematical Analysis and Applications, 38 (1972), 619-632.
doi: 10.1016/0022-247X(72)90073-X. |
| [2] |
R. A. Abrams and A. Ben-Israel,
Nonlinear programming in complex space: necessary conditions, SIAM Journal on Control, 9 (1971), 606-620.
|
| [3] |
A. Aubry, V. Carotenuto and A. De Maio,
New results on generalized fractional programming problems with toeplitz quadratics, IEEE Signal Processing Letters, 23 (2016), 848-852.
|
| [4] |
A. Aubry, A. De Maio, Y. Huang and M. Piezzo,
Robust design of radar doppler filters, IEEE Transactions on Signal Processing, 64 (2016), 5848-5860.
doi: 10.1109/TSP.2016.2576423. |
| [5] |
T. Baleshan, A. Jayalath and J. Coetzee,
On power minimisation and snr maximisation of distributed beamforming in cooperative communication systems, Electronics Letters, 50 (2014), 712-713.
|
| [6] |
O. Besson,
Adaptive detection with bounded steering vectors mismatch angle, IEEE Transactions on Signal Processing, 55 (2007), 1560-1564.
doi: 10.1109/TSP.2006.890820. |
| [7] |
H. Cai, Y. Wang and T. Yi,
An approach for minimizing a quadratically constrained fractional quadratic problem with application to the communications over wireless channels, Optimization Methods and Software, 29 (2014), 310-320.
|
| [8] |
H. Chen, S. Shahbazpanahi and A. B. Gershman,
Filter-and-forward distributed beamforming for two-way relay networks with frequency selective channels, IEEE Transactions on Signal Processing, 60 (2011), 1927-1941.
doi: 10.1109/TSP.2011.2178842. |
| [9] |
J. Chen, H. Lai and S. Schaible,
Complex fractional programming and the charnes-cooper transformation, Journal of Optimization Theory and Applications, 126 (2005), 203-213.
doi: 10.1007/s10957-005-2669-y. |
| [10] |
A. De Maio, Y. Huang, D. P. Palomar, S. Zhang and A. Farina,
Fractional qcqp with applications in ml steering direction estimation for radar detection, IEEE Transactions on Signal Processing, 59 (2010), 172-185.
doi: 10.1109/TSP.2010.2087327. |
| [11] |
W. Dinkelbach,
On nonlinear fractional programming, Management Science, 13 (1967), 492-498.
doi: 10.1287/mnsc.13.7.492. |
| [12] |
S. Fallahi and M. Salahi,
On the complex fractional quadratic optimization with a quadratic constraint, Opsearch, 54 (2017), 94-106.
doi: 10.1007/s12597-016-0263-8. |
| [13] |
M. Grant and S. Boyd, CVX: Matlab Software for Disciplined Convex Programming, Version 2.1, 2014.
doi: 10.1145/2700586. |
| [14] |
J. Hsiao,
On the optimization of mti clutter rejection, IEEE Transactions on Aerospace and Electronic Systems, AES-10 (1974), 622-629.
|
| [15] |
Y. Huang and S. Zhang,
Complex matrix decomposition and quadratic programming, Mathematics of Operations Research, 32 (2007), 758-768.
doi: 10.1287/moor.1070.0268. |
| [16] |
R. Jagannathan,
On some properties of programming problems in parametric form pertaining to fractional programming, Management Science, 12 (1966), 609-615.
doi: 10.1287/mnsc.12.7.609. |
| [17] |
V. Jeyakumar, N. Q. Huy and G. Li, Necessary and sufficient conditions for s-lemma and nonconvex quadratic optimization, Optimization and Engineering, 10 (2009), 491.
doi: 10.1007/s11081-008-9076-9. |
| [18] |
U. T. Jönsson,
A lecture on the s-procedure, Lecture Note at the Royal Institute of Technology, Sweden, 23 (2001), 34-36.
|
| [19] |
N. Levinson,
Linear programming in complex space, Journal of Mathematical Analysis and Applications, 14 (1966), 44-62.
doi: 10.1016/0022-247X(66)90061-8. |
| [20] |
V.-B. Nguyen, R.-L. Sheu and Y. Xia,
An SDP approach for quadratic fractional problems with a two-sided quadratic constraint, Optimization Methods and Software, 31 (2016), 701-719.
doi: 10.1080/10556788.2015.1029575. |
| [21] |
I. Pólik and T. Terlaky,
A survey of the s-lemma, SIAM Review, 49 (2007), 371-418.
doi: 10.1137/S003614450444614X. |
| [22] |
M. Salahi and A. Zare,
SDO relaxation approach to fractional quadratic minimization with one quadratic constraint, Journal of Mathematical Modeling, 3 (2015), 1-13.
|
| [23] |
R. L. Sheu, An SDP approach for some quadratic fractional problems (nonlinear analysis and convex analysis). |
| [24] |
J. F. Sturm and S. Zhang,
On cones of nonnegative quadratic functions, Mathematics of Operations Research, 28 (2003), 246-267.
doi: 10.1287/moor.28.2.246.14485. |
| [25] |
K. Swarup and J. Sharma, Programming with linear fractional functionals in complex spaces, Cahiers du centre Études et de Recherche Opérationelle, 12 (1970), 103–109. |
| [26] |
T. Yi, L. Guo, K. Niu, H. Cai, J. Lin and W. Ai, Cooperative beamforming in cognitive radio network with hybrid relay, in 2012 19th International Conference on Telecommunications (ICT), IEEE, (2012), 1–5. |
| [27] |
A. Zhang and S. Hayashi,
Celis-Dennis-Tapia based approach to quadratic fractional programming problems with two quadratic constraints, Numer. Algebra, Control and Optim., 1 (2011), 83-98.
doi: 10.3934/naco.2011.1.83. |
| [28] |
X. Zhang, Z. He, X. Zhang and W. Peng,
High-performance beampattern synthesis via linear fractional semidefinite relaxation and quasi-convex optimization, IEEE Transactions on Antennas and Propagation, 66 (2018), 3421-3431.
|
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 | |||
| 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 | — | ||||
| 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 | |||
| 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 | — | ||||
| 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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