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Quadratic optimization with two ball constraints
Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran |
In this paper, the minimization of a general quadratic function subject to two ball constraints, called two ball trust-region subproblem (TBTRS), is studied. It is shown that the global optimal solution can be found by solving two extended trust-region subproblems. Strong duality conditions for two special cases are discussed. Finally, a comparison of results of the new algorithm with the other two recently proposed algorithms and CVX software are presented for several classes of randomly generated test problems.
References:
[1] |
S. Adachi, S. Iwata, Y. Nakatsukasa and A. Takeda,
Solving the trust-region subproblem by a generalized eigenvalue problem, SIAM Journal on Optimization, 27 (2017), 269-291.
doi: 10.1137/16M1058200. |
[2] |
S. Ansary Karbasy and M. Salahi, A hybrid algorithm for the two-trust-region subproblem, Computational and Applied Mathematics, https://doi.org/10.1007/s40314-019-0864-y
doi: 10.1007/s40314-019-0864-y. |
[3] |
H. T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Systems, Springer Science and Business Media, 2012.
doi: 10.1007/978-1-4612-3700-6. |
[4] |
A. Beck and Y. C. Eldar,
Strong duality in nonconvex quadratic optimization with two quadratic constraints, SIAM Journal on Optimization, 17 (2006), 844-860.
doi: 10.1137/050644471. |
[5] |
Im. Bomze and Ml. Overton,
Narrowing the difficulty gap for the Celis-Dennis-Tapia problem, Mathematical Programming, 151 (2015), 459-476.
doi: 10.1007/s10107-014-0836-3. |
[6] |
S. Burer and K. M. Anstreicher,
Second-order-cone constraints for extended trust-region subproblems, SIAM Journal on Optimization, 23 (2013), 432-451.
doi: 10.1137/110826862. |
[7] |
S. Burer and B. Yang,
The trust-region subproblem with non-intersecting linear constraints, Mathematical Programming, 149 (2015), 253-264.
doi: 10.1007/s10107-014-0749-1. |
[8] |
M. R. Celis, J. E. Dennis and R. A. Tapia, A trust-region strategy for nonlinear equality constrained optimization, Numerical Optimization, (1984), 71–82. |
[9] |
A. R. Conn, N. I. Gould and P. L. Toint, Trust-Region Methods, SIAM Philadelphia, Vol: 1, 2000.
doi: 10.1137/1.9780898719857. |
[10] |
J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM publisher, Vol: 16, 1996.
doi: 10.1137/1.9781611971200. |
[11] |
S. Fallahi, M. Salahi and S. Ansary Karbasy, On SOCP/SDP formulation of the extended trust-region subproblem, to appear in Iranian Journal of Operations Research, 2019.
doi: 10.1007/s40314-019-0864-y. |
[12] |
M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 2.1, March 2017. Available from: http://cvxr.com/cvx. |
[13] |
S. Gugercin, A. C. Antoulas and H. P. Zhang,
An approach to identification for robust control, IEEE Transactions on Automatic Control, 48 (2003), 1109-1115.
doi: 10.1109/TAC.2003.812821. |
[14] |
M. Heinkenschloss, Mesh independence for nonlinear least squares problems with norm constraints, SlAM Journal on Optimization, 3 (1993), 81-117.
doi: 10.1137/0803005. |
[15] |
M. Heinkenschloss,
On the solution of a two ball trust-region subproblem, Mathematical Programming, 64 (1994), 249-276.
doi: 10.1007/BF01582576. |
[16] |
V. Jeyakumar and G. Y. Li,
Trust-region problems with linear inequality constraints: Exact SDP relaxation,
global optimality and robust optimization, Mathematical Programming, 147 (2014), 171-206.
doi: 10.1007/s10107-013-0716-2. |
[17] |
B. Kaltenbacher, F. Rendl and E. Resmerita,
Computing quasisolutions of nonlinear inverse problems via efficient minimization of trust region problems, Journal of Inverse and Ill-Posed Problems, 24 (2016), 435-447.
doi: 10.1515/jiip-2015-0087. |
[18] |
C. Kravaris and J. H. Seinfeld,
Identification of parameters in distributed parameter systems by regularization, SIAM Journal on Control and Optimization, 23 (1985), 217-241.
doi: 10.1137/0323017. |
[19] |
G. Kristensson, Inverse problems for acoustic waves using the penalised likelihood method, Inverse Problems, 2 (1986), 461. |
[20] |
J. M. Martínez,
Local minimizers of quadratic functions on Euclidean balls and spheres, SIAM Journal on Optimization, 4 (1994), 159-176.
doi: 10.1137/0804009. |
[21] |
Y. Nesterov, H. Wolkowicz and Y. Ye, Semidefinite Programming Relaxations of Nonconvex Quadratic Optimization, Handbook of Semidefinite Programming, Springer, Boston, (2000), 361–419.
doi: 10.1007/978-1-4615-4381-7_13. |
[22] |
S. Omatu and J. H. Seinfeld, Distributed Parameter Systems: Theory and Applications,, Clarendon Press, 1989.
![]() ![]() |
[23] |
F. O'Sullivan and G. Wahba,
A cross validated Bayesian retrieval algorithm for nonlinear remote sensing experiments, Journal of Computational Physics, 59 (1985), 441-455.
|
[24] |
J.-M. Peng and Y. Yuan,
Optimality conditions for the minimization of a quadratic with two quadratic constraints, SIAM Journal on Optimization, 7 (1997), 579-594.
doi: 10.1137/S1052623494261520. |
[25] |
S. Sakaue, Y. Nakatsukasa, A. Takeda and S. Iwata,
Solving generalized CDT problems via two-parameter eigenvalues, SIAM Journal on Optimization, 26 (2016), 1669-1694.
doi: 10.1137/15100624X. |
[26] |
M. Salahi and A. Taati,
A fast eigenvalue approach for solving the trust-region subproblem with an additional linear inequality, Computational and Applied Mathematics, 37 (2018), 329-347.
doi: 10.1007/s40314-016-0347-3. |
[27] |
M. Salahi, A. Taati and H. Wolkowicz,
Local nonglobal minima for solving large scale extended trust-region subproblems, Computational Optimization and Applications, 66 (2016), 223-244.
doi: 10.1007/s10589-016-9867-4. |
[28] |
M. Salah and S. Fallahi,
Trust-region subproblem with an additional linear inequality constraint, Optimization Letters, 10 (2016), 821-832.
doi: 10.1007/s11590-015-0957-5. |
[29] |
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. |
[30] |
C. R. Vogel,
A constrained least squares regularization method for nonlinear iii-posed problems, SIAM Journal on Control and Optimization, 28 (1990), 34-49.
doi: 10.1137/0328002. |
[31] |
A. Zhang and S. Hayashi,
Celis-Dennis-Tapia based approach to quadratic fractional programming problems with two quadratic constraints, Numerical Algebra, Control and Optimization, 1 (2011), 83-98.
doi: 10.3934/naco.2011.1.83. |
show all references
References:
[1] |
S. Adachi, S. Iwata, Y. Nakatsukasa and A. Takeda,
Solving the trust-region subproblem by a generalized eigenvalue problem, SIAM Journal on Optimization, 27 (2017), 269-291.
doi: 10.1137/16M1058200. |
[2] |
S. Ansary Karbasy and M. Salahi, A hybrid algorithm for the two-trust-region subproblem, Computational and Applied Mathematics, https://doi.org/10.1007/s40314-019-0864-y
doi: 10.1007/s40314-019-0864-y. |
[3] |
H. T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Systems, Springer Science and Business Media, 2012.
doi: 10.1007/978-1-4612-3700-6. |
[4] |
A. Beck and Y. C. Eldar,
Strong duality in nonconvex quadratic optimization with two quadratic constraints, SIAM Journal on Optimization, 17 (2006), 844-860.
doi: 10.1137/050644471. |
[5] |
Im. Bomze and Ml. Overton,
Narrowing the difficulty gap for the Celis-Dennis-Tapia problem, Mathematical Programming, 151 (2015), 459-476.
doi: 10.1007/s10107-014-0836-3. |
[6] |
S. Burer and K. M. Anstreicher,
Second-order-cone constraints for extended trust-region subproblems, SIAM Journal on Optimization, 23 (2013), 432-451.
doi: 10.1137/110826862. |
[7] |
S. Burer and B. Yang,
The trust-region subproblem with non-intersecting linear constraints, Mathematical Programming, 149 (2015), 253-264.
doi: 10.1007/s10107-014-0749-1. |
[8] |
M. R. Celis, J. E. Dennis and R. A. Tapia, A trust-region strategy for nonlinear equality constrained optimization, Numerical Optimization, (1984), 71–82. |
[9] |
A. R. Conn, N. I. Gould and P. L. Toint, Trust-Region Methods, SIAM Philadelphia, Vol: 1, 2000.
doi: 10.1137/1.9780898719857. |
[10] |
J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM publisher, Vol: 16, 1996.
doi: 10.1137/1.9781611971200. |
[11] |
S. Fallahi, M. Salahi and S. Ansary Karbasy, On SOCP/SDP formulation of the extended trust-region subproblem, to appear in Iranian Journal of Operations Research, 2019.
doi: 10.1007/s40314-019-0864-y. |
[12] |
M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 2.1, March 2017. Available from: http://cvxr.com/cvx. |
[13] |
S. Gugercin, A. C. Antoulas and H. P. Zhang,
An approach to identification for robust control, IEEE Transactions on Automatic Control, 48 (2003), 1109-1115.
doi: 10.1109/TAC.2003.812821. |
[14] |
M. Heinkenschloss, Mesh independence for nonlinear least squares problems with norm constraints, SlAM Journal on Optimization, 3 (1993), 81-117.
doi: 10.1137/0803005. |
[15] |
M. Heinkenschloss,
On the solution of a two ball trust-region subproblem, Mathematical Programming, 64 (1994), 249-276.
doi: 10.1007/BF01582576. |
[16] |
V. Jeyakumar and G. Y. Li,
Trust-region problems with linear inequality constraints: Exact SDP relaxation,
global optimality and robust optimization, Mathematical Programming, 147 (2014), 171-206.
doi: 10.1007/s10107-013-0716-2. |
[17] |
B. Kaltenbacher, F. Rendl and E. Resmerita,
Computing quasisolutions of nonlinear inverse problems via efficient minimization of trust region problems, Journal of Inverse and Ill-Posed Problems, 24 (2016), 435-447.
doi: 10.1515/jiip-2015-0087. |
[18] |
C. Kravaris and J. H. Seinfeld,
Identification of parameters in distributed parameter systems by regularization, SIAM Journal on Control and Optimization, 23 (1985), 217-241.
doi: 10.1137/0323017. |
[19] |
G. Kristensson, Inverse problems for acoustic waves using the penalised likelihood method, Inverse Problems, 2 (1986), 461. |
[20] |
J. M. Martínez,
Local minimizers of quadratic functions on Euclidean balls and spheres, SIAM Journal on Optimization, 4 (1994), 159-176.
doi: 10.1137/0804009. |
[21] |
Y. Nesterov, H. Wolkowicz and Y. Ye, Semidefinite Programming Relaxations of Nonconvex Quadratic Optimization, Handbook of Semidefinite Programming, Springer, Boston, (2000), 361–419.
doi: 10.1007/978-1-4615-4381-7_13. |
[22] |
S. Omatu and J. H. Seinfeld, Distributed Parameter Systems: Theory and Applications,, Clarendon Press, 1989.
![]() ![]() |
[23] |
F. O'Sullivan and G. Wahba,
A cross validated Bayesian retrieval algorithm for nonlinear remote sensing experiments, Journal of Computational Physics, 59 (1985), 441-455.
|
[24] |
J.-M. Peng and Y. Yuan,
Optimality conditions for the minimization of a quadratic with two quadratic constraints, SIAM Journal on Optimization, 7 (1997), 579-594.
doi: 10.1137/S1052623494261520. |
[25] |
S. Sakaue, Y. Nakatsukasa, A. Takeda and S. Iwata,
Solving generalized CDT problems via two-parameter eigenvalues, SIAM Journal on Optimization, 26 (2016), 1669-1694.
doi: 10.1137/15100624X. |
[26] |
M. Salahi and A. Taati,
A fast eigenvalue approach for solving the trust-region subproblem with an additional linear inequality, Computational and Applied Mathematics, 37 (2018), 329-347.
doi: 10.1007/s40314-016-0347-3. |
[27] |
M. Salahi, A. Taati and H. Wolkowicz,
Local nonglobal minima for solving large scale extended trust-region subproblems, Computational Optimization and Applications, 66 (2016), 223-244.
doi: 10.1007/s10589-016-9867-4. |
[28] |
M. Salah and S. Fallahi,
Trust-region subproblem with an additional linear inequality constraint, Optimization Letters, 10 (2016), 821-832.
doi: 10.1007/s11590-015-0957-5. |
[29] |
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. |
[30] |
C. R. Vogel,
A constrained least squares regularization method for nonlinear iii-posed problems, SIAM Journal on Control and Optimization, 28 (1990), 34-49.
doi: 10.1137/0328002. |
[31] |
A. Zhang and S. Hayashi,
Celis-Dennis-Tapia based approach to quadratic fractional programming problems with two quadratic constraints, Numerical Algebra, Control and Optimization, 1 (2011), 83-98.
doi: 10.3934/naco.2011.1.83. |
Notation | Description |
n | Dimension of problem |
Den | Density of A |
CPU | Run time |
Objective value of the new algorithm | |
Objective value of ADMM algorithm [2] | |
Objective value of SDP relaxation | |
Objective value of Sakaue et. al's algorithm [25] | |
Optimal solution of the new algorithm | |
Optimal solution of Sakaue et. al's algorithm [25] |
Notation | Description |
n | Dimension of problem |
Den | Density of A |
CPU | Run time |
Objective value of the new algorithm | |
Objective value of ADMM algorithm [2] | |
Objective value of SDP relaxation | |
Objective value of Sakaue et. al's algorithm [25] | |
Optimal solution of the new algorithm | |
Optimal solution of Sakaue et. al's algorithm [25] |
n | Den | CPU(TBTRS) | CPU(ADMM) | |
50 | 1 | 0.23 | 0.36 | -9.31e-11 |
100 | 1 | 0.28 | 0.47 | -1.39 e-10 |
200 | 1 | 0.55 | 0.79 | 2.33e-11 |
300 | 1 | 0.80 | 1.75 | 1.86 e-10 |
400 | 1 | 1.58 | 2.63 | 4.66e-10 |
500 | 1 | 2.72 | 3.99 | 8.73e-10 |
700 | 0.1 | 1.49 | 5.174 | 4.54e-10 |
1000 | 0.1 | 3.27 | 20.64 | 3.49e-10 |
2000 | 0.01 | 2.48 | 7.64 | 2.79e-10 |
5000 | 0.001 | 2.74 | 6.59 | -1.16e-10 |
10000 | 0.0001 | 4.62 | 7.29 | -1.46e-11 |
n | Den | CPU(TBTRS) | CPU(ADMM) | |
50 | 1 | 0.23 | 0.36 | -9.31e-11 |
100 | 1 | 0.28 | 0.47 | -1.39 e-10 |
200 | 1 | 0.55 | 0.79 | 2.33e-11 |
300 | 1 | 0.80 | 1.75 | 1.86 e-10 |
400 | 1 | 1.58 | 2.63 | 4.66e-10 |
500 | 1 | 2.72 | 3.99 | 8.73e-10 |
700 | 0.1 | 1.49 | 5.174 | 4.54e-10 |
1000 | 0.1 | 3.27 | 20.64 | 3.49e-10 |
2000 | 0.01 | 2.48 | 7.64 | 2.79e-10 |
5000 | 0.001 | 2.74 | 6.59 | -1.16e-10 |
10000 | 0.0001 | 4.62 | 7.29 | -1.46e-11 |
n | CPU(TBTRS) | CPU(Sakaue et. al's algorithm [25]) | ||
5 | 0.10 | 0.07 | -2.27e-14 | 3.24e-14 |
10 | 0.12 | 0.46 | -1.71e-11 | 1.45e-13 |
15 | 0.23 | 5.94 | -3.87e-12 | 1.78e-14 |
20 | 0.18 | 33.12 | -1.82e-09 | 8.06e-08 |
25 | 0.23 | 130.96 | -2.96e-11 | 9.50e-14 |
30 | 0.22 | 428.21 | -6.58e-09 | 1.26e-10 |
n | CPU(TBTRS) | CPU(Sakaue et. al's algorithm [25]) | ||
5 | 0.10 | 0.07 | -2.27e-14 | 3.24e-14 |
10 | 0.12 | 0.46 | -1.71e-11 | 1.45e-13 |
15 | 0.23 | 5.94 | -3.87e-12 | 1.78e-14 |
20 | 0.18 | 33.12 | -1.82e-09 | 8.06e-08 |
25 | 0.23 | 130.96 | -2.96e-11 | 9.50e-14 |
30 | 0.22 | 428.21 | -6.58e-09 | 1.26e-10 |
n | Den | CPU(TBTRS) | CPU(ADMM) | |
50 | 1 | 0.091 | 2.46 | -3.69e-08 |
100 | 1 | 0.12 | 3.0022 | -1.63e-07 |
200 | 1 | 0.15 | 4.70 | -3.81e-07 |
300 | 1 | 0.24 | 6.71 | -4.61e-07 |
500 | 1 | 0.68 | 13.5 | -6.15e-07 |
700 | 1 | 1.17 | 21.43 | -2.05e-06 |
1000 | 1 | 2.82 | 47.35 | -1.69e-06 |
2000 | 0.1 | 2.39 | 39.16 | -9.22e-07 |
5000 | 0.01 | 2.96 | 55.92 | -1.61e-06 |
10000 | 0.001 | 3.14 | 132.33 | -4.85e-06 |
n | Den | CPU(TBTRS) | CPU(ADMM) | |
50 | 1 | 0.091 | 2.46 | -3.69e-08 |
100 | 1 | 0.12 | 3.0022 | -1.63e-07 |
200 | 1 | 0.15 | 4.70 | -3.81e-07 |
300 | 1 | 0.24 | 6.71 | -4.61e-07 |
500 | 1 | 0.68 | 13.5 | -6.15e-07 |
700 | 1 | 1.17 | 21.43 | -2.05e-06 |
1000 | 1 | 2.82 | 47.35 | -1.69e-06 |
2000 | 0.1 | 2.39 | 39.16 | -9.22e-07 |
5000 | 0.01 | 2.96 | 55.92 | -1.61e-06 |
10000 | 0.001 | 3.14 | 132.33 | -4.85e-06 |
n | CPU(TBTRS) | CPU(Sakaue et. al's algorithm [25]) | ||
5 | 0.06 | 0.051 | 1.42e-11 | 1.22e-06 |
10 | 0.06 | 0.42 | 2.58e-10 | 3.69e-06 |
15 | 0.12 | 6.18 | -1.72e-10 | 2.89e-06 |
20 | 0.08 | 35.03 | 2.84e-09 | 1.07e-05 |
25 | 0.13 | 139.93 | 2.96e-09 | 1.02e-05 |
30 | 0.14 | 449.08 | 4.73e-08 | 3.45e-05 |
n | CPU(TBTRS) | CPU(Sakaue et. al's algorithm [25]) | ||
5 | 0.06 | 0.051 | 1.42e-11 | 1.22e-06 |
10 | 0.06 | 0.42 | 2.58e-10 | 3.69e-06 |
15 | 0.12 | 6.18 | -1.72e-10 | 2.89e-06 |
20 | 0.08 | 35.03 | 2.84e-09 | 1.07e-05 |
25 | 0.13 | 139.93 | 2.96e-09 | 1.02e-05 |
30 | 0.14 | 449.08 | 4.73e-08 | 3.45e-05 |
n | Den | CPU(TBTRS) | CPU(ADMM) | CPU(CVX) | ||
50 | 1 | 0.09 | 2.04 | 0.97 | 2.30e-09 | -1.58e-05 |
100 | 1 | 0.18 | 2.01 | 2.93 | 1.75e-08 | -7.33e-05 |
200 | 1 | 0.16 | 4.05 | 5.41 | 3.82e-07 | -9.45e-04 |
300 | 1 | 0.24 | 6.16 | 14.39 | 5.71e-07 | -2.23e-4 |
400 | 1 | 0.41 | 10.78 | 38.06 | 1.35e-06 | -3.78e-4 |
500 | 1 | 0.67 | 11.87 | 63.26 | 4.67e-07 | -4.67e-4 |
700 | 0.1 | 0.32 | 8.39 | 132.96 | 1.43e-07 | -2.75e-4 |
800 | 0.1 | 0.39 | 9.02 | 206.75 | 2.42e-07 | -2.904e-4 |
900 | 0.1 | 0.44 | 9.67 | 268.47 | 8.91e-07 | -4.88e-4 |
1000 | 0.1 | 0.52 | 11.21 | 427.06 | 3.29e-07 | -5.71e-4 |
2000 | 0.1 | 2.69 | 39.09 | 5.28e-05 | ||
3000 | 0.1 | 5.27 | 82.52 | 3.85e-05 | ||
5000 | 0.01 | 2.99 | 57.99 | 6.16e-05 | ||
10000 | 0.001 | 2.34 | 121.13 | 7.12e-05 |
n | Den | CPU(TBTRS) | CPU(ADMM) | CPU(CVX) | ||
50 | 1 | 0.09 | 2.04 | 0.97 | 2.30e-09 | -1.58e-05 |
100 | 1 | 0.18 | 2.01 | 2.93 | 1.75e-08 | -7.33e-05 |
200 | 1 | 0.16 | 4.05 | 5.41 | 3.82e-07 | -9.45e-04 |
300 | 1 | 0.24 | 6.16 | 14.39 | 5.71e-07 | -2.23e-4 |
400 | 1 | 0.41 | 10.78 | 38.06 | 1.35e-06 | -3.78e-4 |
500 | 1 | 0.67 | 11.87 | 63.26 | 4.67e-07 | -4.67e-4 |
700 | 0.1 | 0.32 | 8.39 | 132.96 | 1.43e-07 | -2.75e-4 |
800 | 0.1 | 0.39 | 9.02 | 206.75 | 2.42e-07 | -2.904e-4 |
900 | 0.1 | 0.44 | 9.67 | 268.47 | 8.91e-07 | -4.88e-4 |
1000 | 0.1 | 0.52 | 11.21 | 427.06 | 3.29e-07 | -5.71e-4 |
2000 | 0.1 | 2.69 | 39.09 | 5.28e-05 | ||
3000 | 0.1 | 5.27 | 82.52 | 3.85e-05 | ||
5000 | 0.01 | 2.99 | 57.99 | 6.16e-05 | ||
10000 | 0.001 | 2.34 | 121.13 | 7.12e-05 |
n | CPU(TBTRS) | CPU(Sakaue et. al's algorithm [25]) | ||
5 | 0.08 | 0.059 | 1.32e-13 | 9.59e-08 |
10 | 0.06 | 0.36 | 1.86e-09 | 1.08e-05 |
15 | 0.06 | 5.69 | 5.12e-09 | 1.66e-05 |
20 | 0.07 | 32.62 | 1.25e-09 | 6.09e-06 |
25 | 0.14 | 130.46 | 7.24e-10 | 3.75e-06 |
30 | 0.14 | 419.58 | 8.38e-09 | 1.72e-05 |
n | CPU(TBTRS) | CPU(Sakaue et. al's algorithm [25]) | ||
5 | 0.08 | 0.059 | 1.32e-13 | 9.59e-08 |
10 | 0.06 | 0.36 | 1.86e-09 | 1.08e-05 |
15 | 0.06 | 5.69 | 5.12e-09 | 1.66e-05 |
20 | 0.07 | 32.62 | 1.25e-09 | 6.09e-06 |
25 | 0.14 | 130.46 | 7.24e-10 | 3.75e-06 |
30 | 0.14 | 419.58 | 8.38e-09 | 1.72e-05 |
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