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New structural properties of inventory models with Polya frequency distributed demand and fixed setup cost
An optimal trade-off model for portfolio selection with sensitivity of parameters
Department of Mathematics, Shanghai University, Shanghai 200444, China |
In this paper, we propose an optimal trade-off model for portfolio selection with sensitivity of parameters, which are estimated from historical data. Mathematically, the model is a quadratic programming problem, whose objective function contains three terms. The first term is a measurement of risk. And the later two are the maximum and minimum sensitivity, which are non-convex and non-smooth functions and lead to the whole model to be an intractable problem. Then we transform this quadratic programming problem into an unconstrained composite problem equivalently. Furthermore, we develop a modified accelerated gradient (AG) algorithm to solve the unconstrained composite problem. The convergence and the convergence rate of our algorithm are derived. Finally, we perform both the empirical analysis and the numerical experiments. The empirical analysis indicates that the optimal trade-off model results in a stable return with lower risk under the stress test. The numerical experiments demonstrate that the modified AG algorithm outperforms the existed AG algorithm for both CPU time and the iterations, respectively.
References:
[1] |
F.A. Al-Khayyal, C. Larsen and T.V. Voorhis,
A relaxation method for nonconvex quadratically constrained quadratic programs, Journal of Global Optimization, 6 (1995), 215-230.
doi: 10.1007/BF01099462. |
[2] |
C. Audet, P. Hansen, B. Jaumard and G. Savard,
A branch-and-cut algorithm for nonconvex quadratically constrained quadratic programming, Mathematical Programming, 87 (2000), 131-152.
doi: 10.1007/s101079900106. |
[3] |
V. Boginski, S. Butenko and P.M. Pardalos,
Statistical analysis of financial networks, Computational Statistics & Data Analysis, 48 (2005), 431-443.
doi: 10.1016/j.csda.2004.02.004. |
[4] |
V.K. Chopra and W.T. Ziemba,
The effect of errors in means, variances, and covariances on optimal portfolio chocie, Journal of Portfolio Management, 19 (1993), 6-11.
doi: 10.3905/jpm.1993.409440. |
[5] |
X.T. Cui, X.L. Sun and D. Sha,
An empirical study on discrete optimization models for portfolio selection, Journal of Industrial and Management Optimization, 5 (2009), 33-46.
doi: 10.3934/jimo.2009.5.33. |
[6] |
X. T. Cui, Studies on Portfolio Selection Problems with Different Risk Measures and Trading Constraints, O224, Fudan University, 2013. Google Scholar |
[7] |
Z.B. Deng, Y.Q. Bai, S.C. Fang, T. Ye and W.X. Xing,
A branch-and-cut approach to potofolio selection with marginal risk control in a linear cone programming framework, Journal of Systems Science and Systems Engineering, 22 (2013), 385-400.
doi: 10.1007/s11518-013-5234-5. |
[8] |
S. Ghadimi and G. H. Lan, Accelerated gradient methods for nonconvex nonlinear and stochastic programming (accepted), Mathematical Programming, (2015). arXiv: 1310.3787v1 Google Scholar |
[9] |
D. Goldfarb and G. Iyengar,
Rubust portfolio selection problems, Mathematices of Operations Reserch, 28 (2003), 1-38.
doi: 10.1287/moor.28.1.1.14260. |
[10] |
M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 2. 1, http://cvxr.com/cvx/, 2014. Google Scholar |
[11] |
P. Horst, P.M. Pardolos and N.V. Thoai, Introduction to Global Optimization, Kluwer Academic Publishers, 1995.
![]() |
[12] |
R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, 1990.
![]() |
[13] |
V. Kalyagin, A. Koldanov, P. Koldanov and V. Zamaraev, Market graph and markowtiz model, Optimization in Science and Engineering: In Honor of the 60th Birthday of Panos M. Pardalos, Springer Science (2014), 293-306. Google Scholar |
[14] |
G.H. Lan,
An optimal method for stochastic composite optimization, Mathematical Programming, 133 (2012), 365-397.
doi: 10.1007/s10107-010-0434-y. |
[15] |
Q. Li and Y.Q. Bai,
Optimal trade-off portfolio selection between total risk and maximum relative marginal risk (accepted), Optimization Methods & Software, 31 (2016), 681-700.
doi: 10.1080/10556788.2015.1041946. |
[16] |
J. Linderoth,
A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs, Mathematical Programming, 103 (2005), 251-282.
doi: 10.1007/s10107-005-0582-7. |
[17] |
H.M. Markowitz,
Portfolio selection, Journal of Finace, 7 (1952), 77-91.
doi: 10.1111/j.1540-6261.1952.tb01525.x. |
[18] |
Y.E. Nesterov,
A method for unconstrained convex minimization problem with the rate of convergence $\mathcal{O}(1/k^2)$, Doklady AN SSSR, 269 (1983), 543-547.
|
[19] |
Y. E. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, Springer, 2003.
doi: 10.007/978-1-4419-8853-9. |
[20] |
Y.E. Nestrov,
Smooth minimization of non-smooth functions, Mathematical Programming, 103 (2005), 127-152.
doi: 10.1007/s10107-004-0552-5. |
[21] |
U. Raber,
A simplicial branch-and-bound method for solving nonconvex all-quadratic programs, Journal of Global Optimization, 13 (1998), 417-432.
doi: 10.1023/A:1008377529330. |
[22] |
B. Scherer,
Can rubust portfolio optimization help to build better portfolios, Journal of Asset Management, 7 (2007), 374-387.
doi: 10.1057/palgrave.jam.2250049. |
[23] |
X.L. Sun, X.J. Zheng and D. Li,
Recent advances in mathematical programming with semi-continuous variables and cardinality constraint, Journal of the Operations Research Society of China, 1 (2013), 55-77.
doi: 10.1007/s40305-013-0004-0. |
[24] |
Y.F. Sun, A. Grace, K.L. Teo and G.L. Zhou,
Portfolio optimization using a new probabilistic risk measure, Journal of Industrial and Management Optimization, 11 (2015), 1275-1283.
doi: 10.3934/jimo.2015.11.1275. |
[25] |
K.L. Teo and X.Q. Yang,
Portfolio selection problem with minimax type risk function, Annals of Operations Research, 101 (2001), 333-349.
doi: 10.1023/A:1010909632198. |
[26] |
Y. Tian, S.C. Fang, Z.B. Deng and Q.W. Jin,
Cardinality constrained portfolio selection problem: A completely positive programming approach, Journal of Industrial and Management Optimization, 12 (2016), 1041-1056.
doi: 10.3934/jimo.2016.12.1041. |
[27] |
S.S. Zhu, D. Li and X.L. Sun,
Portfolio selection with marginal risk control, The Journal of Computational Finance, 14 (2010), 3-28.
doi: 10.21314/JCF.2010.213. |
show all references
References:
[1] |
F.A. Al-Khayyal, C. Larsen and T.V. Voorhis,
A relaxation method for nonconvex quadratically constrained quadratic programs, Journal of Global Optimization, 6 (1995), 215-230.
doi: 10.1007/BF01099462. |
[2] |
C. Audet, P. Hansen, B. Jaumard and G. Savard,
A branch-and-cut algorithm for nonconvex quadratically constrained quadratic programming, Mathematical Programming, 87 (2000), 131-152.
doi: 10.1007/s101079900106. |
[3] |
V. Boginski, S. Butenko and P.M. Pardalos,
Statistical analysis of financial networks, Computational Statistics & Data Analysis, 48 (2005), 431-443.
doi: 10.1016/j.csda.2004.02.004. |
[4] |
V.K. Chopra and W.T. Ziemba,
The effect of errors in means, variances, and covariances on optimal portfolio chocie, Journal of Portfolio Management, 19 (1993), 6-11.
doi: 10.3905/jpm.1993.409440. |
[5] |
X.T. Cui, X.L. Sun and D. Sha,
An empirical study on discrete optimization models for portfolio selection, Journal of Industrial and Management Optimization, 5 (2009), 33-46.
doi: 10.3934/jimo.2009.5.33. |
[6] |
X. T. Cui, Studies on Portfolio Selection Problems with Different Risk Measures and Trading Constraints, O224, Fudan University, 2013. Google Scholar |
[7] |
Z.B. Deng, Y.Q. Bai, S.C. Fang, T. Ye and W.X. Xing,
A branch-and-cut approach to potofolio selection with marginal risk control in a linear cone programming framework, Journal of Systems Science and Systems Engineering, 22 (2013), 385-400.
doi: 10.1007/s11518-013-5234-5. |
[8] |
S. Ghadimi and G. H. Lan, Accelerated gradient methods for nonconvex nonlinear and stochastic programming (accepted), Mathematical Programming, (2015). arXiv: 1310.3787v1 Google Scholar |
[9] |
D. Goldfarb and G. Iyengar,
Rubust portfolio selection problems, Mathematices of Operations Reserch, 28 (2003), 1-38.
doi: 10.1287/moor.28.1.1.14260. |
[10] |
M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 2. 1, http://cvxr.com/cvx/, 2014. Google Scholar |
[11] |
P. Horst, P.M. Pardolos and N.V. Thoai, Introduction to Global Optimization, Kluwer Academic Publishers, 1995.
![]() |
[12] |
R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, 1990.
![]() |
[13] |
V. Kalyagin, A. Koldanov, P. Koldanov and V. Zamaraev, Market graph and markowtiz model, Optimization in Science and Engineering: In Honor of the 60th Birthday of Panos M. Pardalos, Springer Science (2014), 293-306. Google Scholar |
[14] |
G.H. Lan,
An optimal method for stochastic composite optimization, Mathematical Programming, 133 (2012), 365-397.
doi: 10.1007/s10107-010-0434-y. |
[15] |
Q. Li and Y.Q. Bai,
Optimal trade-off portfolio selection between total risk and maximum relative marginal risk (accepted), Optimization Methods & Software, 31 (2016), 681-700.
doi: 10.1080/10556788.2015.1041946. |
[16] |
J. Linderoth,
A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs, Mathematical Programming, 103 (2005), 251-282.
doi: 10.1007/s10107-005-0582-7. |
[17] |
H.M. Markowitz,
Portfolio selection, Journal of Finace, 7 (1952), 77-91.
doi: 10.1111/j.1540-6261.1952.tb01525.x. |
[18] |
Y.E. Nesterov,
A method for unconstrained convex minimization problem with the rate of convergence $\mathcal{O}(1/k^2)$, Doklady AN SSSR, 269 (1983), 543-547.
|
[19] |
Y. E. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, Springer, 2003.
doi: 10.007/978-1-4419-8853-9. |
[20] |
Y.E. Nestrov,
Smooth minimization of non-smooth functions, Mathematical Programming, 103 (2005), 127-152.
doi: 10.1007/s10107-004-0552-5. |
[21] |
U. Raber,
A simplicial branch-and-bound method for solving nonconvex all-quadratic programs, Journal of Global Optimization, 13 (1998), 417-432.
doi: 10.1023/A:1008377529330. |
[22] |
B. Scherer,
Can rubust portfolio optimization help to build better portfolios, Journal of Asset Management, 7 (2007), 374-387.
doi: 10.1057/palgrave.jam.2250049. |
[23] |
X.L. Sun, X.J. Zheng and D. Li,
Recent advances in mathematical programming with semi-continuous variables and cardinality constraint, Journal of the Operations Research Society of China, 1 (2013), 55-77.
doi: 10.1007/s40305-013-0004-0. |
[24] |
Y.F. Sun, A. Grace, K.L. Teo and G.L. Zhou,
Portfolio optimization using a new probabilistic risk measure, Journal of Industrial and Management Optimization, 11 (2015), 1275-1283.
doi: 10.3934/jimo.2015.11.1275. |
[25] |
K.L. Teo and X.Q. Yang,
Portfolio selection problem with minimax type risk function, Annals of Operations Research, 101 (2001), 333-349.
doi: 10.1023/A:1010909632198. |
[26] |
Y. Tian, S.C. Fang, Z.B. Deng and Q.W. Jin,
Cardinality constrained portfolio selection problem: A completely positive programming approach, Journal of Industrial and Management Optimization, 12 (2016), 1041-1056.
doi: 10.3934/jimo.2016.12.1041. |
[27] |
S.S. Zhu, D. Li and X.L. Sun,
Portfolio selection with marginal risk control, The Journal of Computational Finance, 14 (2010), 3-28.
doi: 10.21314/JCF.2010.213. |




Algorithm 2.1. The AG algorithm [8] |
Step 0. Input |
Step 1. Let |
|
Step 2.Compute |
Step 3. If |
Algorithm 2.1. The AG algorithm [8] |
Step 0. Input |
Step 1. Let |
|
Step 2.Compute |
Step 3. If |
Algorithm 4.1. The modified AG algorithm |
Step 0. Input a feasible solution |
Step 1. Let |
|
Step 2. Compute |
Step 3. If |
Algorithm 4.1. The modified AG algorithm |
Step 0. Input a feasible solution |
Step 1. Let |
|
Step 2. Compute |
Step 3. If |
Stock | Name | Stock | Name |
1 | MASTERCARD | 18 | MEDCO HEALTH SLTN. |
2 | PRICELINE.COM | 19 | SOUTHWESTERN ENERGY |
3 | MCDONALDS | 20 | CVS CAREMARK |
4 | AUTOZONE | 21 | J M SMUCKER |
5 | WATSON PHARMS. | 22 | URBAN OUTFITTERS |
6 | FAMILY DOLLAR STORES | 23 | APOLLO GP.'A' |
7 | PERRIGO | 24 | CELGENE |
8 | STERICYCLE | 25 | INTERCONTINENTAL EX. |
9 | INTUITIVE SURGICAL | 26 | LOCKHEED MARTIN |
10 | EDWARDS LIFESCIENCES | 27 | ALTRIA GROUP |
11 | GOODRICH | 28 | HORMEL FOODS |
12 | FIDELITY NAT.INFO.SVS. | 29 | NETFLIX |
13 | F5 NETWORKS | 30 | MICRON TECHNOLOGY |
14 | WAL MART STORES | 31 | VARIAN MED.SYS. |
15 | COCA COLA | 32 | BROWN-FORMAN 'B' |
16 | BIOGEN IDEC | 33 | GAMESTOP 'A' |
17 | TRAVELERS COS. |
Stock | Name | Stock | Name |
1 | MASTERCARD | 18 | MEDCO HEALTH SLTN. |
2 | PRICELINE.COM | 19 | SOUTHWESTERN ENERGY |
3 | MCDONALDS | 20 | CVS CAREMARK |
4 | AUTOZONE | 21 | J M SMUCKER |
5 | WATSON PHARMS. | 22 | URBAN OUTFITTERS |
6 | FAMILY DOLLAR STORES | 23 | APOLLO GP.'A' |
7 | PERRIGO | 24 | CELGENE |
8 | STERICYCLE | 25 | INTERCONTINENTAL EX. |
9 | INTUITIVE SURGICAL | 26 | LOCKHEED MARTIN |
10 | EDWARDS LIFESCIENCES | 27 | ALTRIA GROUP |
11 | GOODRICH | 28 | HORMEL FOODS |
12 | FIDELITY NAT.INFO.SVS. | 29 | NETFLIX |
13 | F5 NETWORKS | 30 | MICRON TECHNOLOGY |
14 | WAL MART STORES | 31 | VARIAN MED.SYS. |
15 | COCA COLA | 32 | BROWN-FORMAN 'B' |
16 | BIOGEN IDEC | 33 | GAMESTOP 'A' |
17 | TRAVELERS COS. |
Expected return(×10−3) | |||||
mean | mean | mean | |||
standard deviation | standard deviation | standard deviation | |||
MV | 0.2050 | 0.2007 | 0.1959 | ||
TMV | 0.2175 | 0.2122 | 0.2036 | ||
TMV | 0.2134 | 0.2055 | 0.1986 | ||
TMV | 0.2089 | 0.2013 | 0.1956 | ||
Optimal portfolio | 0.2225 | 0.2127 | 0.2006 |
Expected return(×10−3) | |||||
mean | mean | mean | |||
standard deviation | standard deviation | standard deviation | |||
MV | 0.2050 | 0.2007 | 0.1959 | ||
TMV | 0.2175 | 0.2122 | 0.2036 | ||
TMV | 0.2134 | 0.2055 | 0.1986 | ||
TMV | 0.2089 | 0.2013 | 0.1956 | ||
Optimal portfolio | 0.2225 | 0.2127 | 0.2006 |
Expected return(×10−3) | |||||
mean | mean | mean | |||
standard deviation | standard deviation | standard deviation | |||
MV | 0.1910 | 0.1877 | 0.1832 | ||
TMV | 0.2016 | 0.1986 | 0.1923 | ||
Optimal portfolio | 0.2066 | 0.1967 | 0.1874 |
Expected return(×10−3) | |||||
mean | mean | mean | |||
standard deviation | standard deviation | standard deviation | |||
MV | 0.1910 | 0.1877 | 0.1832 | ||
TMV | 0.2016 | 0.1986 | 0.1923 | ||
Optimal portfolio | 0.2066 | 0.1967 | 0.1874 |
ρ (×10−3) | τ | Rate (×10−3) | RateR (%) | Sensitivity (×10−3) | RateS (%) |
MV | 0.4086 | - | 3.9889 | - | |
0.4902 | 19.95 | 1.7136 | 57.04 | ||
7 | 0.4276 | 4.65 | 1.9490 | 51.14 | |
0.4210 | 3.02 | 2.0277 | 49.17 | ||
0.4186 | 2.43 | 2.0697 | 48.11 | ||
MV | 0.5145 | - | 4.3323 | - | |
0.5908 | 14.84 | 2.4956 | 42.40 | ||
8 | 0.5444 | 5.81 | 2.8153 | 35.02 | |
0.5351 | 4.01 | 2.9232 | 32.53 | ||
0.5305 | 3.11 | 3.0025 | 30.70 | ||
MV | 0.6511 | - | 6.1201 | - | |
0.7345 | 12.82 | 3.8247 | 37.51 | ||
9 | 0.6852 | 5.25 | 4.1689 | 31.88 | |
0.6758 | 3.80 | 4.2829 | 30.02 | ||
0.6726 | 3.31 | 4.3406 | 29.08 |
ρ (×10−3) | τ | Rate (×10−3) | RateR (%) | Sensitivity (×10−3) | RateS (%) |
MV | 0.4086 | - | 3.9889 | - | |
0.4902 | 19.95 | 1.7136 | 57.04 | ||
7 | 0.4276 | 4.65 | 1.9490 | 51.14 | |
0.4210 | 3.02 | 2.0277 | 49.17 | ||
0.4186 | 2.43 | 2.0697 | 48.11 | ||
MV | 0.5145 | - | 4.3323 | - | |
0.5908 | 14.84 | 2.4956 | 42.40 | ||
8 | 0.5444 | 5.81 | 2.8153 | 35.02 | |
0.5351 | 4.01 | 2.9232 | 32.53 | ||
0.5305 | 3.11 | 3.0025 | 30.70 | ||
MV | 0.6511 | - | 6.1201 | - | |
0.7345 | 12.82 | 3.8247 | 37.51 | ||
9 | 0.6852 | 5.25 | 4.1689 | 31.88 | |
0.6758 | 3.80 | 4.2829 | 30.02 | ||
0.6726 | 3.31 | 4.3406 | 29.08 |
ρ (×10−3) | τ | Rate (×10−3) | RateR (%) | Sensitivity (×10−3) | RateS (%) |
MV | 0.2469 | - | 7.9021 | - | |
0.2819 | 14.18 | 4.8713 | 38.35 | ||
3 | 0.2616 | 5.95 | 5.4896 | 30.53 | |
0.2595 | 5.10 | 5.6577 | 28.40 | ||
0.2582 | 4.58 | 5.4896 | 26.64 | ||
MV | 0.3118 | - | 8.2937 | - | |
0.3505 | 12.41 | 5.386 | 35.06 | ||
3.5 | 0.3356 | 7.63 | 5.6569 | 31.79 | |
0.3308 | 6.09 | 5.9154 | 28.68 | ||
0.3272 | 4.94 | 6.1602 | 25.72 | ||
MV | 0.4445 | - | 9.7782 | - | |
0.4500 | 1.24 | 9.1207 | 6.72 | ||
4 | 0.4482 | 0.83 | 9.2027 | 5.89 | |
0.4472 | 0.61 | 9.2756 | 5.14 | ||
0.4462 | 0.40 | 9.3756 | 4.43 |
ρ (×10−3) | τ | Rate (×10−3) | RateR (%) | Sensitivity (×10−3) | RateS (%) |
MV | 0.2469 | - | 7.9021 | - | |
0.2819 | 14.18 | 4.8713 | 38.35 | ||
3 | 0.2616 | 5.95 | 5.4896 | 30.53 | |
0.2595 | 5.10 | 5.6577 | 28.40 | ||
0.2582 | 4.58 | 5.4896 | 26.64 | ||
MV | 0.3118 | - | 8.2937 | - | |
0.3505 | 12.41 | 5.386 | 35.06 | ||
3.5 | 0.3356 | 7.63 | 5.6569 | 31.79 | |
0.3308 | 6.09 | 5.9154 | 28.68 | ||
0.3272 | 4.94 | 6.1602 | 25.72 | ||
MV | 0.4445 | - | 9.7782 | - | |
0.4500 | 1.24 | 9.1207 | 6.72 | ||
4 | 0.4482 | 0.83 | 9.2027 | 5.89 | |
0.4472 | 0.61 | 9.2756 | 5.14 | ||
0.4462 | 0.40 | 9.3756 | 4.43 |
Opt | Algorithm 4.1 | Algorithm 2.1 | ||||
CPU time | CPU time | |||||
5 | 5.5 | 0.0171 | 96.9 | 58 | 670.6 | 214 |
5 | 6 | 0.0170 | 151.0 | 88 | ≥ 1000 | ≥ 279 |
5 | 7 | 0.0304 | 142.5 | 85 | ≥ 1000 | ≥ 300 |
10 | 5.5 | 0.0297 | 104.2 | 62 | 279.0 | 82 |
10 | 6 | 0.0312 | 123.8 | 75 | 302.2 | 88 |
10 | 7 | 0.0424 | 80.2 | 43 | ≥ 1000 | ≥ 327 |
20 | 5.5 | 0.0351 | 173.0 | 102 | 627.1 | 186 |
20 | 6 | 0.0366 | 141.0 | 84 | 350.5 | 105 |
20 | 7 | 0.0494 | 92.1 | 52 | ≥ 1000 | ≥ 350 |
Opt | Algorithm 4.1 | Algorithm 2.1 | ||||
CPU time | CPU time | |||||
5 | 5.5 | 0.0171 | 96.9 | 58 | 670.6 | 214 |
5 | 6 | 0.0170 | 151.0 | 88 | ≥ 1000 | ≥ 279 |
5 | 7 | 0.0304 | 142.5 | 85 | ≥ 1000 | ≥ 300 |
10 | 5.5 | 0.0297 | 104.2 | 62 | 279.0 | 82 |
10 | 6 | 0.0312 | 123.8 | 75 | 302.2 | 88 |
10 | 7 | 0.0424 | 80.2 | 43 | ≥ 1000 | ≥ 327 |
20 | 5.5 | 0.0351 | 173.0 | 102 | 627.1 | 186 |
20 | 6 | 0.0366 | 141.0 | 84 | 350.5 | 105 |
20 | 7 | 0.0494 | 92.1 | 52 | ≥ 1000 | ≥ 350 |
n | m | min | max | average | Niter |
10 | 5 | 8.2 | 27.5 | 17.6 | 33 |
10 | 10 | 11.5 | 36.0 | 24.4 | 32 |
20 | 5 | 15.5 | 44.8 | 34.1 | 62 |
20 | 10 | 34.6 | 34.9 | 42.6 | 56 |
30 | 10 | 23.4 | 95.5 | 62.1 | 78 |
30 | 20 | 70.0 | 125.3 | 104.1 | 89 |
40 | 10 | 51.9 | 100.1 | 68.4 | 84 |
40 | 20 | 74.8 | 123.5 | 85.3 | 68 |
50 | 10 | 41.8 | 59.4 | 52.0 | 64 |
50 | 20 | 76.3 | 123.0 | 95.5 | 82 |
100 | 10 | 39.6 | 128.1 | 82.0 | 79 |
100 | 20 | 85.8 | 277.5 | 156.7 | 71 |
100 | 50 | 615.8 | 830.8 | 738.2 | 185 |
200 | 20 | 225.5 | 530.3 | 405.3 | 137 |
200 | 50 | 722.9 | 1106.8 | 911.8 | 165 |
n | m | min | max | average | Niter |
10 | 5 | 8.2 | 27.5 | 17.6 | 33 |
10 | 10 | 11.5 | 36.0 | 24.4 | 32 |
20 | 5 | 15.5 | 44.8 | 34.1 | 62 |
20 | 10 | 34.6 | 34.9 | 42.6 | 56 |
30 | 10 | 23.4 | 95.5 | 62.1 | 78 |
30 | 20 | 70.0 | 125.3 | 104.1 | 89 |
40 | 10 | 51.9 | 100.1 | 68.4 | 84 |
40 | 20 | 74.8 | 123.5 | 85.3 | 68 |
50 | 10 | 41.8 | 59.4 | 52.0 | 64 |
50 | 20 | 76.3 | 123.0 | 95.5 | 82 |
100 | 10 | 39.6 | 128.1 | 82.0 | 79 |
100 | 20 | 85.8 | 277.5 | 156.7 | 71 |
100 | 50 | 615.8 | 830.8 | 738.2 | 185 |
200 | 20 | 225.5 | 530.3 | 405.3 | 137 |
200 | 50 | 722.9 | 1106.8 | 911.8 | 165 |
n | m | min | max | average | Niter |
10 | 5 | 12.8 | 14.5 | 13.3 | 25 |
10 | 10 | 36.7 | 37.4 | 36.9 | 51 |
20 | 5 | 7.1 | 7.4 | 7.3 | 14 |
20 | 10 | 14.6 | 34.3 | 24.8 | 32 |
30 | 10 | 13.2 | 15.5 | 14.5 | 20 |
30 | 20 | 45.4 | 46.9 | 46.1 | 39 |
40 | 10 | 9.4 | 10.0 | 9.7 | 13 |
40 | 20 | 22.7 | 26.1 | 24.0 | 19 |
50 | 10 | 8.6 | 10.4 | 9.5 | 13 |
50 | 20 | 22.3 | 24.2 | 23.1 | 19 |
100 | 10 | 5.8 | 12.8 | 9.3 | 10 |
100 | 20 | 16.6 | 28.8 | 22.6 | 13 |
100 | 50 | 47.1 | 54.9 | 51.5 | 15 |
200 | 20 | 33.3 | 39.6 | 36.5 | 17 |
200 | 50 | 50.9 | 78.1 | 66.7 | 14 |
n | m | min | max | average | Niter |
10 | 5 | 12.8 | 14.5 | 13.3 | 25 |
10 | 10 | 36.7 | 37.4 | 36.9 | 51 |
20 | 5 | 7.1 | 7.4 | 7.3 | 14 |
20 | 10 | 14.6 | 34.3 | 24.8 | 32 |
30 | 10 | 13.2 | 15.5 | 14.5 | 20 |
30 | 20 | 45.4 | 46.9 | 46.1 | 39 |
40 | 10 | 9.4 | 10.0 | 9.7 | 13 |
40 | 20 | 22.7 | 26.1 | 24.0 | 19 |
50 | 10 | 8.6 | 10.4 | 9.5 | 13 |
50 | 20 | 22.3 | 24.2 | 23.1 | 19 |
100 | 10 | 5.8 | 12.8 | 9.3 | 10 |
100 | 20 | 16.6 | 28.8 | 22.6 | 13 |
100 | 50 | 47.1 | 54.9 | 51.5 | 15 |
200 | 20 | 33.3 | 39.6 | 36.5 | 17 |
200 | 50 | 50.9 | 78.1 | 66.7 | 14 |
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