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January  2023, 19(1): 30-55. doi: 10.3934/jimo.2021173

Projection method with inertial step for nonlinear equations: Application to signal recovery

1. 

KMUTTFixed Point Research Laboratory, Room SCL 802, Fixed Point Laboratory, Science Laboratory Building, Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand

2. 

Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand

3. 

Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan

4. 

School of Mathematics and Statistics, Zaozhuang University, Shandong 277160, China

5. 

School of Management, Qufu Normal University, Shandong 276826, China

6. 

NCAO Research Center, Fixed Point Theory and Applications Research Group, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand

7. 

Department of Mathematical Sciences, Faculty of Physical Sciences, Bayero University, Kano. Kano, Nigeria, Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, Pretoria, Medunsa-0204, South Africa

* Corresponding author: Poom Kumam

Received  May 2021 Revised  August 2021 Published  January 2023 Early access  October 2021

Fund Project: The first author is supported by the Petchra Pra Jom Klao Ph.D. Research Scholarship from King Mongkut's University of Technology Thonburi (Grant no. 16/2561)

In this paper, using the concept of inertial extrapolation, we introduce a globally convergent inertial extrapolation method for solving nonlinear equations with convex constraints for which the underlying mapping is monotone and Lipschitz continuous. The method can be viewed as a combination of the efficient three-term derivative-free method of Gao and He [Calcolo. 55(4), 1-17, 2018] with the inertial extrapolation step. Moreover, the algorithm is designed such that at every iteration, the method is free from derivative evaluations. Under standard assumptions, we establish the global convergence results for the proposed method. Numerical implementations illustrate the performance and advantage of this new method. Moreover, we also extend this method to solve the LASSO problems to decode a sparse signal in compressive sensing. Performance comparisons illustrate the effectiveness and competitiveness of our algorithm.

Citation: Abdulkarim Hassan Ibrahim, Poom Kumam, Min Sun, Parin Chaipunya, Auwal Bala Abubakar. Projection method with inertial step for nonlinear equations: Application to signal recovery. Journal of Industrial and Management Optimization, 2023, 19 (1) : 30-55. doi: 10.3934/jimo.2021173
References:
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A. B. AbubakarA. H. IbrahimA. B. Muhammad and C. Tammer, A modified descent Dai-Yuan conjugate gradient method for constraint nonlinear monotone operator equations, Appl. Anal. Optim., 4 (2020), 1-24. 

[2]

A. B. Abubakar and P. Kumam, A descent Dai-Liao conjugate gradient method for nonlinear equations, Numerical Algorithms, 81 (2019), 197-210.  doi: 10.1007/s11075-018-0541-z.

[3]

A. B. Abubakar and P. Kumam, An improved three-term derivative-free method for solving nonlinear equations, Comput. Appl. Math., 37 (2018), 6760-6773.  doi: 10.1007/s40314-018-0712-5.

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A. B. Abubakar, P. Kumam and A. H. Ibrahim, Inertial Derivative-Free Projection Method for Nonlinear Monotone Operator Equations with Convex Constraints, IEEE Access. 2021.

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J. AbubakarP. KumamA. H. Ibrahim and et al., Inertial iterative schemes with variable step sizes for variational inequality problem involving pseudomonotone operator, Mathematics, 8 (2020), 609. 

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A. B. Abubakar, P. Kumam, A. H. Ibrahim, P. Chaipunya and S. A. Rano, New Hybrid Three-Term Spectral-Conjugate Gradient Method for Finding Solutions of Nonlinear Monotone Operator Equations with Applications, Mathematics and Computers in Simulation, 2021.

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A. B. Abubakar, P. Kumam, A. H. Ibrahim and J. Rilwan, Derivative-free HS-DY-type method for solving nonlinear equations and image restoration, Heliyon, 6 (2020), e05400.

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A. B. Abubakar, P. Kumam and H. Mohammad, A note on the spectral gradient projection method for nonlinear monotone equations with applications, Comput. Appl. Math., 39 (2020), Paper No. 129, 35 pp. doi: 10.1007/s40314-020-01151-5.

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A. B. Abubakar, K. Muangchoo, A. H. Ibrahim, S. E. Fadugba, K. O. Aremu and L. O. Jolaoso, A modified scaled spectral-conjugate gradient-based algorithm for solving monotone operator equations, J. Math., 2021 (2021), Art. ID 5549878, 9 pp. doi: 10.1155/2021/5549878.

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J. AbubakarK. SombutH. ur RehmanA. H. Ibrahim and et al., An accelerated subgradient extragradient algorithm for strongly pseudomonotone variational inequality problems, Thai J. Math., 18 (2020), 166-187. 

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A. B. Abubakar, P. Kumam, M. Malik and A. H. Ibrahim, A Hybrid Conjugate Gradient Based Approach for Solving Unconstrained Optimization and Motion Control Problems., Mathematics and Computers in Simulation, 2021.

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R. I. Boţ and E. R. Csetnek, An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems, Numer. Algorithms, 71 (2016), 519-540.  doi: 10.1007/s11075-015-0007-5.

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R. I. Boţ and E. R. Csetnek, A hybrid proximal-extragradient algorithm with inertial effects, Numer. Funct. Anal. Optim., 36 (2015), 951-963.  doi: 10.1080/01630563.2015.1042113.

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R. I. BoţE. R. Csetnek and C. Hendrich, Inertial Douglas–Rachford splitting for monotone inclusion problems, Appl. Math. Comput., 256 (2015), 472-487.  doi: 10.1016/j.amc.2015.01.017.

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Q. L. DongY. J. ChoL. L. Zhong and Th. M. Rassias, Inertial projection and contraction algorithms for variational inequalities, J. Global Optim., 70 (2018), 687-704.  doi: 10.1007/s10898-017-0506-0.

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P. Gao and C. He, An efficient three-term conjugate gradient method for nonlinear monotone equations with convex constraints, Calcolo, 55 (2018), Paper No. 53, 17 pp. doi: 10.1007/s10092-018-0291-2.

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A. H. Ibrahim, J. Deepho, A. B. Abubakar and A. Adamu, A three-term Polak-Ribiére-Polyak derivative-free method and its application to image restoration, Scientific African, 13 (2021), e00880. Available from: https://www.sciencedirect.com/science/article/pii/S2468227621001848.

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A. H. Ibrahim, J. Deepho, A. B. Abubakar and K. O. Aremu, A Modified Liu-Storey-Conjugate Descent Hybrid Projection Method for Convex Constrained Nonlinear Equations and Image Restoration, Numerical Algebra, Control & Optimization, 2021.

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A. H. Ibrahim and P. Kumam, Re-modified derivative-free iterative method for nonlinear monotone equations with convex constraints, Ain Shams Engineering Journal, (2021).

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A. H. IbrahimP. KumamA. B. AbubakarU. Batsari YusufS. E. Yimer and K. O. Aremu, An efficient gradient-free projection algorithm for constrained nonlinear equations and image restoration, AIMS Math., 6 (2021), 235-260.  doi: 10.3934/math.2021016.

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A. H. Ibrahim, P. Kumam, A. B. Abubakar, W. Jirakitpuwapat and J. Abubakar, A hybrid conjugate gradient algorithm for constrained monotone equations with application in compressive sensing, Heliyon, 6 (2020), e03466.

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show all references

References:
[1]

A. B. AbubakarA. H. IbrahimA. B. Muhammad and C. Tammer, A modified descent Dai-Yuan conjugate gradient method for constraint nonlinear monotone operator equations, Appl. Anal. Optim., 4 (2020), 1-24. 

[2]

A. B. Abubakar and P. Kumam, A descent Dai-Liao conjugate gradient method for nonlinear equations, Numerical Algorithms, 81 (2019), 197-210.  doi: 10.1007/s11075-018-0541-z.

[3]

A. B. Abubakar and P. Kumam, An improved three-term derivative-free method for solving nonlinear equations, Comput. Appl. Math., 37 (2018), 6760-6773.  doi: 10.1007/s40314-018-0712-5.

[4]

A. B. Abubakar, P. Kumam and A. H. Ibrahim, Inertial Derivative-Free Projection Method for Nonlinear Monotone Operator Equations with Convex Constraints, IEEE Access. 2021.

[5]

J. AbubakarP. KumamA. H. Ibrahim and et al., Inertial iterative schemes with variable step sizes for variational inequality problem involving pseudomonotone operator, Mathematics, 8 (2020), 609. 

[6]

A. B. Abubakar, P. Kumam, A. H. Ibrahim, P. Chaipunya and S. A. Rano, New Hybrid Three-Term Spectral-Conjugate Gradient Method for Finding Solutions of Nonlinear Monotone Operator Equations with Applications, Mathematics and Computers in Simulation, 2021.

[7]

A. B. Abubakar, P. Kumam, A. H. Ibrahim and J. Rilwan, Derivative-free HS-DY-type method for solving nonlinear equations and image restoration, Heliyon, 6 (2020), e05400.

[8]

A. B. Abubakar, P. Kumam and H. Mohammad, A note on the spectral gradient projection method for nonlinear monotone equations with applications, Comput. Appl. Math., 39 (2020), Paper No. 129, 35 pp. doi: 10.1007/s40314-020-01151-5.

[9]

A. B. AbubakarP. KumamH. Mohammad and A. H. Ibrahim, PRP-like algorithm for monotone operator equations, Jpn. J. Ind. Appl. Math., 38 (2021), 805-822.  doi: 10.1007/s13160-021-00462-2.

[10]

A. B. AbubakarK. MuangchooA. H. IbrahimJ. Abubakar and S. A. Rano, FR-type algorithm for finding approximate solutions to nonlinear monotone operator equations, Arab. J. Math. (Springer), 10 (2021), 261-270.  doi: 10.1007/s40065-021-00313-5.

[11]

A. B. Abubakar, K. Muangchoo, A. H. Ibrahim, S. E. Fadugba, K. O. Aremu and L. O. Jolaoso, A modified scaled spectral-conjugate gradient-based algorithm for solving monotone operator equations, J. Math., 2021 (2021), Art. ID 5549878, 9 pp. doi: 10.1155/2021/5549878.

[12]

A. B. AbubakarK. MuangchooA. H. IbrahimA. B. MuhammadL. O. Jolaoso and K. O. Aremu, A new three-term Hestenes-Stiefel type method for nonlinear monotone operator equations and image restoration, IEEE Access, 9 (2021), 18262-18277. 

[13]

A. B. AbubakarJ. RilwanS. E. YimerA. H. Ibrahim and and I. Ahmed, Spectral three-term conjugate descent method for solving nonlinear monotone equations with convex constraints, Thai J. Math., 18 (2020), 501-517. 

[14]

J. AbubakarP. KumamA. H. Ibrahim and A. Padcharoen, Relaxed inertial Tseng's type method for solving the inclusion problem with application to image restoration, Mathematics, 8 (2020), 818. 

[15]

J. AbubakarK. SombutH. ur RehmanA. H. Ibrahim and et al., An accelerated subgradient extragradient algorithm for strongly pseudomonotone variational inequality problems, Thai J. Math., 18 (2020), 166-187. 

[16]

W. Aj and B. Wollenberg, Power Generation, Operation and Control, New York: John Wiley & Sons. 1996,592.

[17]

F. Alvarez and H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Analysis, 9 (2001), 3-11.  doi: 10.1023/A:1011253113155.

[18]

H. AttouchJ. Peypouquet and P. Redont, A dynamical approach to an inertial forward-backward algorithm for convex minimization, SIAM J. Optim., 24 (2014), 232-256.  doi: 10.1137/130910294.

[19]

A. AuslenderM. Teboulle and S. Ben-Tiba, A logarithmic-quadratic proximal method for variational inequalities, Comput. Optim. Appl., 12 (1999), 31-40.  doi: 10.1023/A:1008607511915.

[20]

A. B. AbubakarP. KumamM. MalikP. Chaipunya and A. H. Ibrahim, A hybrid FR-DY conjugate gradient algorithm for unconstrained optimization with application in portfolio selection, AIMS Math., 6 (2021), 6506-6527.  doi: 10.3934/math.2021383.

[21]

A. B. Abubakar, P. Kumam, M. Malik and A. H. Ibrahim, A Hybrid Conjugate Gradient Based Approach for Solving Unconstrained Optimization and Motion Control Problems., Mathematics and Computers in Simulation, 2021.

[22]

R. I. Boţ and E. R. Csetnek, An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems, Numer. Algorithms, 71 (2016), 519-540.  doi: 10.1007/s11075-015-0007-5.

[23]

R. I. Boţ and E. R. Csetnek, A hybrid proximal-extragradient algorithm with inertial effects, Numer. Funct. Anal. Optim., 36 (2015), 951-963.  doi: 10.1080/01630563.2015.1042113.

[24]

R. I. BoţE. R. Csetnek and C. Hendrich, Inertial Douglas–Rachford splitting for monotone inclusion problems, Appl. Math. Comput., 256 (2015), 472-487.  doi: 10.1016/j.amc.2015.01.017.

[25]

C. ChenR. H. ChanS. Ma and J. Yang, Inertial proximal ADMM for linearly constrained separable convex optimization, SIAM J. Imaging Sci., 8 (2015), 2239-2267.  doi: 10.1137/15100463X.

[26]

P. Chuasuk, F. Ogbuisi, Y. Shehu and P. Cholamjiak, New inertial method for generalized split variational inclusion problems, Journal of Industrial & Management Optimization, (2020).

[27]

J. E. Dennis Jr. and J. J. Moré, A characterization of superlinear convergence and its application to quasi-Newton methods, Math. Comp., 28 (1974), 549-560.  doi: 10.1090/S0025-5718-1974-0343581-1.

[28]

J. E. Dennis Jr. and J. J. Moré, Quasi-Newton methods, motivation and theory, SIAM Rev., 19 (1977), 46-89.  doi: 10.1137/1019005.

[29]

J. E. Dennis Jr. and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM; 1996. doi: 10.1137/1.9781611971200.

[30]

S. P. Dirkse and M. C. Ferris, MCPLIB: A collection of nonlinear mixed complementarity problems, Optimization Methods and Software, 5 (1995), 319-345. 

[31]

E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles, Math. Program., 91 (2002), 201-213.  doi: 10.1007/s101070100263.

[32]

Q. L. DongY. J. ChoL. L. Zhong and Th. M. Rassias, Inertial projection and contraction algorithms for variational inequalities, J. Global Optim., 70 (2018), 687-704.  doi: 10.1007/s10898-017-0506-0.

[33]

M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing, Springer Science & Business Media; 2010. doi: 10.1007/978-1-4419-7011-4.

[34]

M. A. FigueiredoR. D. Nowak and S. J. Wright, Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems, IEEE Journal of Selected Topics in Signal Processing, 1 (2007), 586-597. 

[35]

P. Gao and C. He, An efficient three-term conjugate gradient method for nonlinear monotone equations with convex constraints, Calcolo, 55 (2018), Paper No. 53, 17 pp. doi: 10.1007/s10092-018-0291-2.

[36]

A. H. Ibrahim, J. Deepho, A. B. Abubakar and A. Adamu, A three-term Polak-Ribiére-Polyak derivative-free method and its application to image restoration, Scientific African, 13 (2021), e00880. Available from: https://www.sciencedirect.com/science/article/pii/S2468227621001848.

[37]

A. H. Ibrahim, J. Deepho, A. B. Abubakar and K. O. Aremu, A Modified Liu-Storey-Conjugate Descent Hybrid Projection Method for Convex Constrained Nonlinear Equations and Image Restoration, Numerical Algebra, Control & Optimization, 2021.

[38]

A. H. IbrahimG. A. IsaH. UsmanJ. Abubakar and A. B. Abubakar, Derivative-free RMIL conjugate gradient method for convex constrained equations, Thai J. Math., 18 (2019), 212-232. 

[39]

A. H. Ibrahim and P. Kumam, Re-modified derivative-free iterative method for nonlinear monotone equations with convex constraints, Ain Shams Engineering Journal, (2021).

[40]

A. H. IbrahimP. KumamA. B. AbubakarJ. Abubakar and A. B. Muhammad, Least-square-based three-term conjugate gradient projection method for $\ell_1$-norm problems with application to compressed sensing, Mathematics, 8 (2020), 602. 

[41]

A. H. IbrahimP. KumamA. B. AbubakarU. Batsari YusufS. E. Yimer and K. O. Aremu, An efficient gradient-free projection algorithm for constrained nonlinear equations and image restoration, AIMS Math., 6 (2021), 235-260.  doi: 10.3934/math.2021016.

[42]

A. H. Ibrahim, P. Kumam, A. B. Abubakar, W. Jirakitpuwapat and J. Abubakar, A hybrid conjugate gradient algorithm for constrained monotone equations with application in compressive sensing, Heliyon, 6 (2020), e03466.

[43]

A. H. IbrahimP. KumamA. B. AbubakarU. B. Yusuf and J. Rilwan, Derivative-free conjugate residual algorithms for convex constraints nonlinear monotone equations and signal recovery, J. Nonlinear Convex Anal., 21 (2020), 1959-1972. 

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Figure 1.  Performance profile on iteration (Iner.Algo versus Algo)
Figure 2.  Performance profile on function evaluations (Iner.Algo versus Algo)
Figure 3.  Performance profile on CPU time (Iner.Algo versus Algo)
Figure 4.  Numerical results of Problem (36) with $ \delta = 0 $
Figure 5.  Numerical results of Problem (36) with $ \delta = 10^{-4} $
Figure 6.  Numerical results of Problem (36) with $ \delta = 10^{-3} $
Figure 7.  Numerical results of Problem (36) with $ \delta = 10^{-2} $
Table 1.  Numerical results of Problem (36) with different noise
Standard deviation Methods RelErr Iter Times (s)
$ \delta $=0 Iner.Algo 0.0317 279 2.15
Algo 0.0317 468 3.15
$ \delta=10^{-4} $ Iner.Algo 0.0281 292 2.21
Algo 0.0281 593 4.22
$ \delta=10^{-3} $ Iner.Algo 0.0328 288 2.39
Algo 0.0241 438 2.96
$ \delta=10^{-2} $ Iner.Algo 0.0518 337 2.55
Algo 0.0518 939 6.75
Standard deviation Methods RelErr Iter Times (s)
$ \delta $=0 Iner.Algo 0.0317 279 2.15
Algo 0.0317 468 3.15
$ \delta=10^{-4} $ Iner.Algo 0.0281 292 2.21
Algo 0.0281 593 4.22
$ \delta=10^{-3} $ Iner.Algo 0.0328 288 2.39
Algo 0.0241 438 2.96
$ \delta=10^{-2} $ Iner.Algo 0.0518 337 2.55
Algo 0.0518 939 6.75
Table 2.  Test results for problem 1
INER. ALGO ALGO.
DIM INP NI NF CPU NORM NI NF CPU NORM
1000 Case I 6 24 0.006537 4.32E-08 11 44 0.009092 5.72E-07
Case II 7 28 0.008529 1.46E-07 11 44 0.013711 9.90E-07
Case III 7 28 0.006432 1.93E-07 12 48 0.013874 2.73E-07
Case IV 8 32 0.010414 1.79E-07 13 52 0.022209 3.00E-07
Case V 6 24 0.010836 7.10E-07 13 52 0.013571 6.02E-07
Case VI 8 32 0.00719 1.96E-07 13 52 0.010269 2.99E-07
Case VII 11 44 0.008426 1.12E-07 13 52 0.013525 6.09E-07
5000 Case I 6 24 0.22912 2.95E-08 12 48 0.10538 2.54E-07
Case II 7 28 0.85665 1.13E-07 12 48 0.079825 4.39E-07
Case III 7 28 0.53427 4.32E-07 12 48 0.032032 5.68E-07
Case IV 8 32 0.059319 3.98E-07 13 52 0.044075 4.28E-07
Case V 7 28 0.038857 3.61E-08 13 52 0.044154 4.83E-07
Case VI 8 32 0.065865 4.39E-07 13 52 0.18386 3.91E-07
Case VII 11 44 0.045807 3.86E-07 14 56 0.055375 2.68E-07
10000 Case I 6 24 0.055883 2.62E-08 12 48 0.26245 3.59E-07
Case II 7 28 0.10776 9.11E-08 12 48 0.084684 6.19E-07
Case III 7 28 0.12502 6.11E-07 12 48 0.07673 7.96E-07
Case IV 8 32 0.060193 5.62E-07 13 52 0.082261 5.60E-07
Case V 7 28 0.0602 5.18E-08 13 52 0.068433 4.87E-07
Case VI 8 32 0.05542 6.21E-07 13 52 0.066654 5.01E-07
Case VII 13 52 0.085591 1.80E-07 14 56 0.13534 3.78E-07
50000 Case I 6 24 0.17095 2.91E-08 12 48 0.23645 8.03E-07
Case II 7 28 0.15714 6.26E-08 13 52 0.25119 2.77E-07
Case III 8 32 0.57011 2.73E-08 13 52 0.21362 3.53E-07
Case IV 9 36 0.46934 2.51E-08 14 56 0.26703 2.34E-07
Case V 7 28 0.29169 1.17E-07 13 52 0.2287 7.20E-07
Case VI 9 36 0.2315 2.78E-08 14 56 0.24843 2.05E-07
Case VII 13 52 0.56004 2.37E-08 14 56 0.32785 8.43E-07
100000 Case I 6 24 0.98438 3.58E-08 13 52 0.43828 2.27E-07
Case II 7 28 0.39255 6.17E-08 13 52 0.62503 3.91E-07
Case III 8 32 0.48543 3.86E-08 13 52 0.42684 4.99E-07
Case IV 9 36 0.52322 3.56E-08 14 56 0.50903 3.28E-07
Case V 7 28 0.43716 1.66E-07 13 52 0.62726 9.52E-07
Case VI 9 36 0.54937 3.93E-08 14 56 0.49271 2.86E-07
Case VII 13 52 0.89929 4.99E-08 15 60 0.53354 2.38E-07
INER. ALGO ALGO.
DIM INP NI NF CPU NORM NI NF CPU NORM
1000 Case I 6 24 0.006537 4.32E-08 11 44 0.009092 5.72E-07
Case II 7 28 0.008529 1.46E-07 11 44 0.013711 9.90E-07
Case III 7 28 0.006432 1.93E-07 12 48 0.013874 2.73E-07
Case IV 8 32 0.010414 1.79E-07 13 52 0.022209 3.00E-07
Case V 6 24 0.010836 7.10E-07 13 52 0.013571 6.02E-07
Case VI 8 32 0.00719 1.96E-07 13 52 0.010269 2.99E-07
Case VII 11 44 0.008426 1.12E-07 13 52 0.013525 6.09E-07
5000 Case I 6 24 0.22912 2.95E-08 12 48 0.10538 2.54E-07
Case II 7 28 0.85665 1.13E-07 12 48 0.079825 4.39E-07
Case III 7 28 0.53427 4.32E-07 12 48 0.032032 5.68E-07
Case IV 8 32 0.059319 3.98E-07 13 52 0.044075 4.28E-07
Case V 7 28 0.038857 3.61E-08 13 52 0.044154 4.83E-07
Case VI 8 32 0.065865 4.39E-07 13 52 0.18386 3.91E-07
Case VII 11 44 0.045807 3.86E-07 14 56 0.055375 2.68E-07
10000 Case I 6 24 0.055883 2.62E-08 12 48 0.26245 3.59E-07
Case II 7 28 0.10776 9.11E-08 12 48 0.084684 6.19E-07
Case III 7 28 0.12502 6.11E-07 12 48 0.07673 7.96E-07
Case IV 8 32 0.060193 5.62E-07 13 52 0.082261 5.60E-07
Case V 7 28 0.0602 5.18E-08 13 52 0.068433 4.87E-07
Case VI 8 32 0.05542 6.21E-07 13 52 0.066654 5.01E-07
Case VII 13 52 0.085591 1.80E-07 14 56 0.13534 3.78E-07
50000 Case I 6 24 0.17095 2.91E-08 12 48 0.23645 8.03E-07
Case II 7 28 0.15714 6.26E-08 13 52 0.25119 2.77E-07
Case III 8 32 0.57011 2.73E-08 13 52 0.21362 3.53E-07
Case IV 9 36 0.46934 2.51E-08 14 56 0.26703 2.34E-07
Case V 7 28 0.29169 1.17E-07 13 52 0.2287 7.20E-07
Case VI 9 36 0.2315 2.78E-08 14 56 0.24843 2.05E-07
Case VII 13 52 0.56004 2.37E-08 14 56 0.32785 8.43E-07
100000 Case I 6 24 0.98438 3.58E-08 13 52 0.43828 2.27E-07
Case II 7 28 0.39255 6.17E-08 13 52 0.62503 3.91E-07
Case III 8 32 0.48543 3.86E-08 13 52 0.42684 4.99E-07
Case IV 9 36 0.52322 3.56E-08 14 56 0.50903 3.28E-07
Case V 7 28 0.43716 1.66E-07 13 52 0.62726 9.52E-07
Case VI 9 36 0.54937 3.93E-08 14 56 0.49271 2.86E-07
Case VII 13 52 0.89929 4.99E-08 15 60 0.53354 2.38E-07
Table 3.  Test results for problem 2
INER. ALGO ALGO.
DIM INP NI NF CPU NORM NI NF CPU NORM
1000 Case I 4 12 0.087572 7.61E-09 4 12 0.008488 5.17E-07
Case II 5 15 0.016259 6.04E-09 5 15 0.006875 6.04E-09
Case III 5 15 0.011455 4.37E-07 5 15 0.006569 4.37E-07
Case IV 6 18 0.011153 1.52E-07 6 18 0.008092 1.52E-07
Case V 7 21 0.00633 1.10E-09 7 21 0.005713 1.10E-09
Case VI 7 21 0.011702 1.74E-08 7 21 0.004815 1.74E-08
Case VII 11 35 0.012359 2.77E-07 15 49 0.021261 3.66E-07
5000 Case I 4 12 0.064074 8.56E-10 4 12 0.035632 1.75E-07
Case II 5 15 0.076987 6.27E-10 5 15 0.019409 6.27E-10
Case III 5 15 0.027949 1.42E-07 5 15 0.014664 1.42E-07
Case IV 6 18 0.036548 3.94E-08 6 18 0.026202 3.94E-08
Case V 6 18 0.01692 4.05E-07 6 18 0.018243 4.05E-07
Case VI 7 21 0.025943 2.36E-09 7 21 0.027663 2.36E-09
Case VII 13 43 0.094345 3.76E-09 18 62 0.19571 1.75E-07
10000 Case I 4 12 0.022826 3.98E-10 4 12 0.02201 1.21E-07
Case II 5 15 0.029782 2.79E-10 5 15 0.03389 2.79E-10
Case III 5 15 0.036108 9.73E-08 5 15 0.020746 9.73E-08
Case IV 6 18 0.031098 2.56E-08 6 18 0.031954 2.56E-08
Case V 6 18 0.11664 2.93E-07 6 18 0.13272 2.93E-07
Case VI 7 21 0.039807 1.24E-09 7 21 0.056343 1.24E-09
Case VII 13 43 0.37509 1.17E-08 20 69 0.12928 9.06E-08
50000 Case I 4 13 0.36887 1.05E-10 4 12 0.15651 6.32E-08
Case II 5 16 0.42862 6.75E-11 5 16 0.10554 6.75E-11
Case III 5 15 0.24321 4.87E-08 5 15 0.16698 4.87E-08
Case IV 6 18 0.46737 1.11E-08 6 18 0.1099 1.11E-08
Case V 6 18 0.13319 1.84E-07 6 18 0.11779 1.84E-07
Case VI 7 21 0.18161 4.01E-10 7 21 0.1203 4.01E-10
Case VII 16 54 0.46124 1.11E-09 23 81 0.5211 6.89E-07
100000 Case I 4 13 0.26038 6.80E-11 4 12 0.13094 5.40E-08
Case II 5 16 0.30277 4.27E-11 5 16 0.18755 4.27E-11
Case III 5 15 0.4347 4.05E-08 5 15 0.27985 4.05E-08
Case IV 6 18 0.42556 8.15E-09 6 18 0.19207 8.15E-09
Case V 6 18 0.25488 1.80E-07 6 18 0.20156 1.80E-07
Case VI 7 21 0.43576 2.71E-10 7 21 0.21005 2.71E-10
Case VII 15 51 0.82806 1.64E-09 24 86 1.1405 4.14E-09
INER. ALGO ALGO.
DIM INP NI NF CPU NORM NI NF CPU NORM
1000 Case I 4 12 0.087572 7.61E-09 4 12 0.008488 5.17E-07
Case II 5 15 0.016259 6.04E-09 5 15 0.006875 6.04E-09
Case III 5 15 0.011455 4.37E-07 5 15 0.006569 4.37E-07
Case IV 6 18 0.011153 1.52E-07 6 18 0.008092 1.52E-07
Case V 7 21 0.00633 1.10E-09 7 21 0.005713 1.10E-09
Case VI 7 21 0.011702 1.74E-08 7 21 0.004815 1.74E-08
Case VII 11 35 0.012359 2.77E-07 15 49 0.021261 3.66E-07
5000 Case I 4 12 0.064074 8.56E-10 4 12 0.035632 1.75E-07
Case II 5 15 0.076987 6.27E-10 5 15 0.019409 6.27E-10
Case III 5 15 0.027949 1.42E-07 5 15 0.014664 1.42E-07
Case IV 6 18 0.036548 3.94E-08 6 18 0.026202 3.94E-08
Case V 6 18 0.01692 4.05E-07 6 18 0.018243 4.05E-07
Case VI 7 21 0.025943 2.36E-09 7 21 0.027663 2.36E-09
Case VII 13 43 0.094345 3.76E-09 18 62 0.19571 1.75E-07
10000 Case I 4 12 0.022826 3.98E-10 4 12 0.02201 1.21E-07
Case II 5 15 0.029782 2.79E-10 5 15 0.03389 2.79E-10
Case III 5 15 0.036108 9.73E-08 5 15 0.020746 9.73E-08
Case IV 6 18 0.031098 2.56E-08 6 18 0.031954 2.56E-08
Case V 6 18 0.11664 2.93E-07 6 18 0.13272 2.93E-07
Case VI 7 21 0.039807 1.24E-09 7 21 0.056343 1.24E-09
Case VII 13 43 0.37509 1.17E-08 20 69 0.12928 9.06E-08
50000 Case I 4 13 0.36887 1.05E-10 4 12 0.15651 6.32E-08
Case II 5 16 0.42862 6.75E-11 5 16 0.10554 6.75E-11
Case III 5 15 0.24321 4.87E-08 5 15 0.16698 4.87E-08
Case IV 6 18 0.46737 1.11E-08 6 18 0.1099 1.11E-08
Case V 6 18 0.13319 1.84E-07 6 18 0.11779 1.84E-07
Case VI 7 21 0.18161 4.01E-10 7 21 0.1203 4.01E-10
Case VII 16 54 0.46124 1.11E-09 23 81 0.5211 6.89E-07
100000 Case I 4 13 0.26038 6.80E-11 4 12 0.13094 5.40E-08
Case II 5 16 0.30277 4.27E-11 5 16 0.18755 4.27E-11
Case III 5 15 0.4347 4.05E-08 5 15 0.27985 4.05E-08
Case IV 6 18 0.42556 8.15E-09 6 18 0.19207 8.15E-09
Case V 6 18 0.25488 1.80E-07 6 18 0.20156 1.80E-07
Case VI 7 21 0.43576 2.71E-10 7 21 0.21005 2.71E-10
Case VII 15 51 0.82806 1.64E-09 24 86 1.1405 4.14E-09
Table 4.  Test results for problem 3
INER. ALGO ALGO.
DIM INP NI NF CPU NORM NI NF CPU NORM
1000 Case I 13 52 0.075605 3.36E-07 31 124 0.023401 6.98E-07
Case II 14 56 0.033438 9.91E-07 32 126 0.016722 2.09E-22
Case III 15 59 0.014509 6.78E-07 34 135 0.029896 7.24E-07
Case IV 15 59 0.022327 8.58E-07 35 140 0.033452 8.69E-07
Case V 15 59 0.012348 4.93E-07 35 140 0.024649 9.73E-07
Case VI 16 64 0.011655 6.63E-07 36 142 0.041312 0
Case VII 15 60 0.019587 7.37E-07 34 136 0.018809 8.10E-07
5000 Case I 13 51 0.084308 7.51E-07 30 118 0.095447 0
Case II 15 60 0.052927 6.65E-07 31 122 0.50476 5.62E-21
Case III 16 62 0.08685 0 35 140 0.084126 9.71E-07
Case IV 16 62 0.054943 1.40E-21 36 143 0.095079 7.25E-21
Case V 16 64 0.10298 3.31E-07 35 138 0.26686 1.40E-21
Case VI 17 66 0.121 2.34E-22 36 142 0.16331 0
Case VII 16 64 0.82826 5.03E-07 36 144 0.16091 6.28E-07
10000 Case I 14 55 0.094948 3.19E-07 31 122 0.12401 6.62E-22
Case II 15 60 0.17919 9.40E-07 32 126 0.14805 0
Case III 16 62 0.085448 3.31E-22 36 143 0.13867 8.24E-07
Case IV 16 64 0.12333 8.14E-07 35 138 0.15826 2.65E-21
Case V 16 62 0.1041 3.31E-22 35 138 0.27449 6.62E-22
Case VI 17 66 0.11764 0 36 143 0.16026 2.65E-21
Case VII 16 64 0.081495 7.03E-07 36 144 0.19776 8.97E-07
50000 Case I 12 46 0.4807 1.15E-19 31 122 0.53862 2.96E-21
Case II 16 62 0.85283 0 31 122 0.5534 2.96E-21
Case III 16 62 0.43053 3.70E-21 33 130 0.66305 0
Case IV 15 58 0.34538 1.78E-20 34 134 0.59923 2.37E-20
Case V 17 66 0.45061 0 35 138 0.82508 1.48E-21
Case VI 18 72 0.46072 4.22E-07 36 143 0.62746 2.96E-21
Case VII 17 68 0.40219 4.73E-07 38 152 0.6653 7.24E-07
100000 Case I 15 60 0.73965 3.02E-07 31 122 1.139 2.09E-21
Case II 16 62 1.1267 5.23E-22 37 148 1.361 6.48E-07
Case III 17 68 0.58078 6.10E-07 38 151 1.5385 9.38E-07
Case IV 17 68 0.64576 7.73E-07 37 146 1.3166 0
Case V 17 68 0.77823 4.44E-07 40 158 1.4507 0
Case VI 17 66 0.67135 3.14E-21 40 159 1.4451 8.14E-07
Case VII 17 68 0.67963 6.70E-07 39 156 1.5115 6.14E-07
INER. ALGO ALGO.
DIM INP NI NF CPU NORM NI NF CPU NORM
1000 Case I 13 52 0.075605 3.36E-07 31 124 0.023401 6.98E-07
Case II 14 56 0.033438 9.91E-07 32 126 0.016722 2.09E-22
Case III 15 59 0.014509 6.78E-07 34 135 0.029896 7.24E-07
Case IV 15 59 0.022327 8.58E-07 35 140 0.033452 8.69E-07
Case V 15 59 0.012348 4.93E-07 35 140 0.024649 9.73E-07
Case VI 16 64 0.011655 6.63E-07 36 142 0.041312 0
Case VII 15 60 0.019587 7.37E-07 34 136 0.018809 8.10E-07
5000 Case I 13 51 0.084308 7.51E-07 30 118 0.095447 0
Case II 15 60 0.052927 6.65E-07 31 122 0.50476 5.62E-21
Case III 16 62 0.08685 0 35 140 0.084126 9.71E-07
Case IV 16 62 0.054943 1.40E-21 36 143 0.095079 7.25E-21
Case V 16 64 0.10298 3.31E-07 35 138 0.26686 1.40E-21
Case VI 17 66 0.121 2.34E-22 36 142 0.16331 0
Case VII 16 64 0.82826 5.03E-07 36 144 0.16091 6.28E-07
10000 Case I 14 55 0.094948 3.19E-07 31 122 0.12401 6.62E-22
Case II 15 60 0.17919 9.40E-07 32 126 0.14805 0
Case III 16 62 0.085448 3.31E-22 36 143 0.13867 8.24E-07
Case IV 16 64 0.12333 8.14E-07 35 138 0.15826 2.65E-21
Case V 16 62 0.1041 3.31E-22 35 138 0.27449 6.62E-22
Case VI 17 66 0.11764 0 36 143 0.16026 2.65E-21
Case VII 16 64 0.081495 7.03E-07 36 144 0.19776 8.97E-07
50000 Case I 12 46 0.4807 1.15E-19 31 122 0.53862 2.96E-21
Case II 16 62 0.85283 0 31 122 0.5534 2.96E-21
Case III 16 62 0.43053 3.70E-21 33 130 0.66305 0
Case IV 15 58 0.34538 1.78E-20 34 134 0.59923 2.37E-20
Case V 17 66 0.45061 0 35 138 0.82508 1.48E-21
Case VI 18 72 0.46072 4.22E-07 36 143 0.62746 2.96E-21
Case VII 17 68 0.40219 4.73E-07 38 152 0.6653 7.24E-07
100000 Case I 15 60 0.73965 3.02E-07 31 122 1.139 2.09E-21
Case II 16 62 1.1267 5.23E-22 37 148 1.361 6.48E-07
Case III 17 68 0.58078 6.10E-07 38 151 1.5385 9.38E-07
Case IV 17 68 0.64576 7.73E-07 37 146 1.3166 0
Case V 17 68 0.77823 4.44E-07 40 158 1.4507 0
Case VI 17 66 0.67135 3.14E-21 40 159 1.4451 8.14E-07
Case VII 17 68 0.67963 6.70E-07 39 156 1.5115 6.14E-07
Table 5.  Test results for problem 4
INER. ALGO ALGO.
DIM INP NI NF CPU NORM NI NF CPU NORM
1000 Case I 13 51 0.06059 3.25E-07 31 124 0.02262 6.43E-07
Case II 14 56 0.008665 7.01E-07 32 128 0.020618 7.10E-07
Case III 14 56 0.023062 7.14E-07 33 132 0.030138 8.25E-07
Case IV 14 56 0.009094 9.53E-07 33 132 0.022234 9.09E-07
Case V 15 60 0.019868 8.47E-07 32 128 0.020698 6.86E-07
Case VI 15 60 0.032655 7.79E-07 35 140 0.033957 6.08E-07
Case VII 16 64 0.016672 4.18E-07 33 132 0.030958 7.44E-07
5000 Case I 13 51 0.057064 7.27E-07 32 127 0.083999 8.63E-07
Case II 15 59 0.047515 4.70E-07 33 132 0.15239 9.52E-07
Case III 15 59 0.032093 4.79E-07 35 140 0.061537 6.64E-07
Case IV 15 59 0.062007 6.39E-07 35 140 0.13051 7.32E-07
Case V 16 63 0.045188 5.68E-07 33 132 0.071242 9.20E-07
Case VI 16 64 0.037473 5.22E-07 36 144 0.060172 8.16E-07
Case VII 16 64 0.030254 8.70E-07 35 140 0.072475 6.05E-07
10000 Case I - - - - 33 131 0.12064 7.32E-07
Case II 15 59 0.048278 6.65E-07 34 135 0.63105 8.08E-07
Case III 15 59 0.057155 6.77E-07 35 140 0.11101 9.39E-07
Case IV 15 60 0.060725 9.04E-07 36 143 0.1951 6.21E-07
Case V 16 63 0.053938 8.04E-07 34 135 0.12425 7.81E-07
Case VI 16 63 0.066593 7.39E-07 37 147 0.20795 6.92E-07
Case VII 17 68 0.057214 3.60E-07 35 140 0.12255 8.50E-07
50000 Case I 14 55 1.0077 6.89E-07 34 134 0.42136 0
Case II 16 63 0.61724 4.46E-07 35 138 0.43246 0
Case III 16 63 0.32509 4.54E-07 37 147 1.0982 7.56E-07
Case IV 16 63 0.19589 6.06E-07 37 148 0.59668 8.33E-07
Case V 17 66 0.24309 0 - - - -
Case VI - - - - - - - -
Case VII 17 68 0.2842 8.32E-07 37 148 0.47886 6.85E-07
100000 Case I 14 54 0.32344 0 - - - -
Case II 16 63 0.51789 6.31E-07 35 138 0.81304 0
Case III 16 62 0.5148 0 38 150 1.2176 0
Case IV - - - - 38 151 1.0824 7.07E-07
Case V 17 68 0.56565 7.63E-07 35 138 1.0007 0
Case VI 17 68 0.57103 7.01E-07 - - - -
Case VII 18 72 0.53447 3.50E-07 37 148 0.88698 9.70E-07
INER. ALGO ALGO.
DIM INP NI NF CPU NORM NI NF CPU NORM
1000 Case I 13 51 0.06059 3.25E-07 31 124 0.02262 6.43E-07
Case II 14 56 0.008665 7.01E-07 32 128 0.020618 7.10E-07
Case III 14 56 0.023062 7.14E-07 33 132 0.030138 8.25E-07
Case IV 14 56 0.009094 9.53E-07 33 132 0.022234 9.09E-07
Case V 15 60 0.019868 8.47E-07 32 128 0.020698 6.86E-07
Case VI 15 60 0.032655 7.79E-07 35 140 0.033957 6.08E-07
Case VII 16 64 0.016672 4.18E-07 33 132 0.030958 7.44E-07
5000 Case I 13 51 0.057064 7.27E-07 32 127 0.083999 8.63E-07
Case II 15 59 0.047515 4.70E-07 33 132 0.15239 9.52E-07
Case III 15 59 0.032093 4.79E-07 35 140 0.061537 6.64E-07
Case IV 15 59 0.062007 6.39E-07 35 140 0.13051 7.32E-07
Case V 16 63 0.045188 5.68E-07 33 132 0.071242 9.20E-07
Case VI 16 64 0.037473 5.22E-07 36 144 0.060172 8.16E-07
Case VII 16 64 0.030254 8.70E-07 35 140 0.072475 6.05E-07
10000 Case I - - - - 33 131 0.12064 7.32E-07
Case II 15 59 0.048278 6.65E-07 34 135 0.63105 8.08E-07
Case III 15 59 0.057155 6.77E-07 35 140 0.11101 9.39E-07
Case IV 15 60 0.060725 9.04E-07 36 143 0.1951 6.21E-07
Case V 16 63 0.053938 8.04E-07 34 135 0.12425 7.81E-07
Case VI 16 63 0.066593 7.39E-07 37 147 0.20795 6.92E-07
Case VII 17 68 0.057214 3.60E-07 35 140 0.12255 8.50E-07
50000 Case I 14 55 1.0077 6.89E-07 34 134 0.42136 0
Case II 16 63 0.61724 4.46E-07 35 138 0.43246 0
Case III 16 63 0.32509 4.54E-07 37 147 1.0982 7.56E-07
Case IV 16 63 0.19589 6.06E-07 37 148 0.59668 8.33E-07
Case V 17 66 0.24309 0 - - - -
Case VI - - - - - - - -
Case VII 17 68 0.2842 8.32E-07 37 148 0.47886 6.85E-07
100000 Case I 14 54 0.32344 0 - - - -
Case II 16 63 0.51789 6.31E-07 35 138 0.81304 0
Case III 16 62 0.5148 0 38 150 1.2176 0
Case IV - - - - 38 151 1.0824 7.07E-07
Case V 17 68 0.56565 7.63E-07 35 138 1.0007 0
Case VI 17 68 0.57103 7.01E-07 - - - -
Case VII 18 72 0.53447 3.50E-07 37 148 0.88698 9.70E-07
Table 6.  Test results for problem 5
INER. ALGO ALGO.
DIM INP NI NF CPU NORM NI NF CPU NORM
1000 Case I 25 95 0.1329 6.45E-07 36 139 0.019135 6.42E-07
Case II 19 72 0.014534 4.04E-07 36 140 0.018994 6.62E-07
Case III 22 86 0.022553 9.29E-07 47 175 0.044522 6.46E-07
Case IV 22 88 0.036427 4.00E-07 40 156 0.03131 6.54E-07
Case V 25 100 0.037173 4.90E-07 37 146 0.023349 7.12E-07
Case VI 41 164 0.071153 6.01E-07 36 143 0.037977 9.25E-07
Case VII 42 167 0.12029 4.55E-07 49 184 0.060357 9.87E-07
5000 Case I 22 83 0.13573 6.53E-07 36 139 0.11871 6.40E-07
Case II 19 72 0.11048 8.54E-07 37 144 0.088934 9.52E-07
Case III 25 98 0.090732 3.03E-07 51 189 0.10334 9.00E-07
Case IV 24 96 0.061785 8.38E-07 44 169 0.44477 9.18E-07
Case V 30 120 0.1862 4.11E-07 40 157 0.11434 6.51E-07
Case VI 54 216 0.33113 9.45E-07 41 160 0.10301 8.14E-07
Case VII 69 275 0.5637 3.29E-07 47 180 0.107 9.84E-07
10000 Case I 27 103 0.14553 4.01E-07 36 139 0.1347 7.33E-07
Case II 20 76 0.086196 3.64E-07 38 147 0.15397 8.23E-07
Case III 25 98 0.20935 8.75E-07 47 177 0.17983 7.72E-07
Case IV 24 96 0.19624 5.28E-07 44 170 0.19929 7.87E-07
Case V 35 140 0.1977 6.41E-07 42 163 0.18619 9.45E-07
Case VI 60 240 0.65802 4.12E-07 49 185 0.47874 7.04E-07
Case VII 84 335 0.96443 8.09E-07 43 169 0.1934 7.22E-07
50000 Case I 55 215 2.6998 9.96E-07 65 255 1.3087 8.37E-07
Case II 20 76 0.2954 8.35E-07 48 179 0.92835 6.84E-07
Case III 27 106 0.47985 8.73E-07 67 240 0.92921 6.31E-07
Case IV 41 164 1.7421 9.71E-07 47 181 0.72302 6.44E-07
Case V 36 144 1.5835 8.46E-07 45 174 0.86921 7.83E-07
Case VI 79 316 4.0512 6.61E-07 54 201 0.94527 9.68E-07
Case VII 103 411 6.5137 7.57E-07 52 204 0.79431 7.54E-07
100000 Case I 74 291 7.1551 8.21E-07 83 327 3.7002 7.04E-07
Case II 21 80 0.671 3.61E-07 46 173 1.4825 9.75E-07
Case III 26 102 0.89984 6.93E-07 55 204 1.6728 8.97E-07
Case IV 44 176 2.6239 8.75E-07 52 196 1.5811 9.15E-07
Case V 43 172 2.2944 8.30E-07 48 184 1.5047 6.71E-07
Case VI 99 396 12.5198 3.58E-07 47 181 1.4726 8.26E-07
Case VII 111 443 12.3882 5.20E-07 69 272 2.5982 7.84E-07
INER. ALGO ALGO.
DIM INP NI NF CPU NORM NI NF CPU NORM
1000 Case I 25 95 0.1329 6.45E-07 36 139 0.019135 6.42E-07
Case II 19 72 0.014534 4.04E-07 36 140 0.018994 6.62E-07
Case III 22 86 0.022553 9.29E-07 47 175 0.044522 6.46E-07
Case IV 22 88 0.036427 4.00E-07 40 156 0.03131 6.54E-07
Case V 25 100 0.037173 4.90E-07 37 146 0.023349 7.12E-07
Case VI 41 164 0.071153 6.01E-07 36 143 0.037977 9.25E-07
Case VII 42 167 0.12029 4.55E-07 49 184 0.060357 9.87E-07
5000 Case I 22 83 0.13573 6.53E-07 36 139 0.11871 6.40E-07
Case II 19 72 0.11048 8.54E-07 37 144 0.088934 9.52E-07
Case III 25 98 0.090732 3.03E-07 51 189 0.10334 9.00E-07
Case IV 24 96 0.061785 8.38E-07 44 169 0.44477 9.18E-07
Case V 30 120 0.1862 4.11E-07 40 157 0.11434 6.51E-07
Case VI 54 216 0.33113 9.45E-07 41 160 0.10301 8.14E-07
Case VII 69 275 0.5637 3.29E-07 47 180 0.107 9.84E-07
10000 Case I 27 103 0.14553 4.01E-07 36 139 0.1347 7.33E-07
Case II 20 76 0.086196 3.64E-07 38 147 0.15397 8.23E-07
Case III 25 98 0.20935 8.75E-07 47 177 0.17983 7.72E-07
Case IV 24 96 0.19624 5.28E-07 44 170 0.19929 7.87E-07
Case V 35 140 0.1977 6.41E-07 42 163 0.18619 9.45E-07
Case VI 60 240 0.65802 4.12E-07 49 185 0.47874 7.04E-07
Case VII 84 335 0.96443 8.09E-07 43 169 0.1934 7.22E-07
50000 Case I 55 215 2.6998 9.96E-07 65 255 1.3087 8.37E-07
Case II 20 76 0.2954 8.35E-07 48 179 0.92835 6.84E-07
Case III 27 106 0.47985 8.73E-07 67 240 0.92921 6.31E-07
Case IV 41 164 1.7421 9.71E-07 47 181 0.72302 6.44E-07
Case V 36 144 1.5835 8.46E-07 45 174 0.86921 7.83E-07
Case VI 79 316 4.0512 6.61E-07 54 201 0.94527 9.68E-07
Case VII 103 411 6.5137 7.57E-07 52 204 0.79431 7.54E-07
100000 Case I 74 291 7.1551 8.21E-07 83 327 3.7002 7.04E-07
Case II 21 80 0.671 3.61E-07 46 173 1.4825 9.75E-07
Case III 26 102 0.89984 6.93E-07 55 204 1.6728 8.97E-07
Case IV 44 176 2.6239 8.75E-07 52 196 1.5811 9.15E-07
Case V 43 172 2.2944 8.30E-07 48 184 1.5047 6.71E-07
Case VI 99 396 12.5198 3.58E-07 47 181 1.4726 8.26E-07
Case VII 111 443 12.3882 5.20E-07 69 272 2.5982 7.84E-07
Table 7.  Test results for problem 6
INER. ALGO ALGO.
DIM INP NI NF CPU NORM NI NF CPU NORM
1000 Case I 17 68 0.070165 3.66E-07 37 148 0.086194 8.53E-07
Case II 17 68 0.019243 3.42E-07 37 148 0.049573 8.20E-07
Case III 17 68 0.022663 3.01E-07 37 148 0.051928 7.22E-07
Case IV 16 64 0.016058 6.87E-07 36 144 0.074625 8.24E-07
Case V 16 64 0.028711 5.52E-07 36 144 0.080704 6.61E-07
Case VI 16 64 0.036264 3.25E-07 35 140 0.098138 6.50E-07
Case VII 17 68 0.029906 3.02E-07 37 148 0.057573 7.28E-07
5000 Case I 17 68 0.11976 8.21E-07 39 156 0.16301 6.87E-07
Case II 17 68 0.13738 7.66E-07 39 156 0.1438 6.61E-07
Case III 17 68 0.070333 6.75E-07 38 152 0.14908 9.71E-07
Case IV 17 68 0.097123 4.62E-07 38 152 0.16691 6.64E-07
Case V 17 68 0.070428 3.71E-07 37 148 0.15817 8.88E-07
Case VI 16 64 0.065808 7.29E-07 36 144 0.36454 8.73E-07
Case VII 17 68 0.089469 6.80E-07 38 152 0.19723 9.77E-07
10000 Case I 18 72 0.1274 3.48E-07 39 156 0.35497 9.72E-07
Case II 18 72 0.12939 3.25E-07 39 156 0.50582 9.35E-07
Case III 17 68 0.13873 9.55E-07 39 156 0.46728 8.24E-07
Case IV 17 68 0.13388 6.54E-07 38 152 0.2839 9.40E-07
Case V 17 68 0.20628 5.24E-07 38 152 0.27388 7.54E-07
Case VI 17 68 0.16002 3.09E-07 37 148 0.56098 7.41E-07
Case VII 17 68 0.16743 9.63E-07 39 156 0.46017 8.28E-07
50000 Case I 18 70 0.74334 0 40 158 1.2103 0
Case II 18 70 0.54759 0 40 158 1.4011 0
Case III 18 70 0.46758 0 39 154 1.3023 0
Case IV 18 70 0.47821 0 39 154 1.2953 0
Case V 18 70 0.51731 0 38 150 1.2347 0
Case VI 17 66 0.52485 0 37 146 1.2817 0
Case VII 18 70 0.55512 0 40 158 1.3476 0
100000 Case I 18 70 1.0765 0 39 154 2.4974 0
Case II 18 70 1.3017 0 39 154 2.5376 0
Case III 18 70 1.2128 0 38 150 2.45 0
Case IV 17 66 1.2676 0 38 150 2.3559 0
Case V 17 66 1.1563 0 37 146 2.4861 0
Case VI 17 66 1.0464 0 36 142 2.2564 0
Case VII 18 70 1.0587 0 39 154 2.6041 0
INER. ALGO ALGO.
DIM INP NI NF CPU NORM NI NF CPU NORM
1000 Case I 17 68 0.070165 3.66E-07 37 148 0.086194 8.53E-07
Case II 17 68 0.019243 3.42E-07 37 148 0.049573 8.20E-07
Case III 17 68 0.022663 3.01E-07 37 148 0.051928 7.22E-07
Case IV 16 64 0.016058 6.87E-07 36 144 0.074625 8.24E-07
Case V 16 64 0.028711 5.52E-07 36 144 0.080704 6.61E-07
Case VI 16 64 0.036264 3.25E-07 35 140 0.098138 6.50E-07
Case VII 17 68 0.029906 3.02E-07 37 148 0.057573 7.28E-07
5000 Case I 17 68 0.11976 8.21E-07 39 156 0.16301 6.87E-07
Case II 17 68 0.13738 7.66E-07 39 156 0.1438 6.61E-07
Case III 17 68 0.070333 6.75E-07 38 152 0.14908 9.71E-07
Case IV 17 68 0.097123 4.62E-07 38 152 0.16691 6.64E-07
Case V 17 68 0.070428 3.71E-07 37 148 0.15817 8.88E-07
Case VI 16 64 0.065808 7.29E-07 36 144 0.36454 8.73E-07
Case VII 17 68 0.089469 6.80E-07 38 152 0.19723 9.77E-07
10000 Case I 18 72 0.1274 3.48E-07 39 156 0.35497 9.72E-07
Case II 18 72 0.12939 3.25E-07 39 156 0.50582 9.35E-07
Case III 17 68 0.13873 9.55E-07 39 156 0.46728 8.24E-07
Case IV 17 68 0.13388 6.54E-07 38 152 0.2839 9.40E-07
Case V 17 68 0.20628 5.24E-07 38 152 0.27388 7.54E-07
Case VI 17 68 0.16002 3.09E-07 37 148 0.56098 7.41E-07
Case VII 17 68 0.16743 9.63E-07 39 156 0.46017 8.28E-07
50000 Case I 18 70 0.74334 0 40 158 1.2103 0
Case II 18 70 0.54759 0 40 158 1.4011 0
Case III 18 70 0.46758 0 39 154 1.3023 0
Case IV 18 70 0.47821 0 39 154 1.2953 0
Case V 18 70 0.51731 0 38 150 1.2347 0
Case VI 17 66 0.52485 0 37 146 1.2817 0
Case VII 18 70 0.55512 0 40 158 1.3476 0
100000 Case I 18 70 1.0765 0 39 154 2.4974 0
Case II 18 70 1.3017 0 39 154 2.5376 0
Case III 18 70 1.2128 0 38 150 2.45 0
Case IV 17 66 1.2676 0 38 150 2.3559 0
Case V 17 66 1.1563 0 37 146 2.4861 0
Case VI 17 66 1.0464 0 36 142 2.2564 0
Case VII 18 70 1.0587 0 39 154 2.6041 0
Table 8.  Test results for problem 7
INER. ALGO ALGO.
DIM INP NI NF CPU NORM NI NF CPU NORM
1000 Case I 9 36 0.029847 1.25E-07 14 56 0.010049 4.58E-07
Case II 8 32 0.01614 7.06E-07 14 56 0.009322 3.19E-07
Case III 7 28 0.036217 2.03E-07 11 44 0.014851 6.44E-07
Case IV 8 32 0.00896 1.86E-07 15 60 0.009756 2.52E-07
Case V 9 36 0.012299 2.91E-07 15 60 0.014388 3.91E-07
Case VI 9 35 0.012932 3.19E-07 15 59 0.01453 2.82E-07
Case VII 9 36 0.015981 3.43E-07 14 56 0.017579 3.00E-07
5000 Case I 9 36 0.040232 2.79E-07 15 60 0.06212 2.57E-07
Case II 9 36 0.038549 1.30E-07 14 56 0.035553 7.13E-07
Case III 7 28 0.02036 4.53E-07 12 48 0.037788 3.61E-07
Case IV 8 32 0.025654 4.15E-07 15 60 0.035425 5.64E-07
Case V 9 36 0.026546 6.51E-07 15 60 0.12771 8.73E-07
Case VI 9 35 0.096229 7.14E-07 15 59 0.044662 6.31E-07
Case VII 9 36 0.093227 7.84E-07 14 56 0.053641 6.71E-07
10000 Case I 9 36 0.054105 3.95E-07 15 60 0.055328 3.64E-07
Case II 9 36 0.10633 1.84E-07 15 60 0.10212 2.53E-07
Case III 7 28 0.037857 6.41E-07 12 48 0.066777 5.11E-07
Case IV 8 32 0.082416 5.87E-07 15 60 0.071481 7.98E-07
Case V 9 36 0.18582 9.20E-07 16 64 0.059125 3.10E-07
Case VI 10 39 0.070373 8.34E-08 15 59 0.081851 8.93E-07
Case VII 10 40 0.071963 9.12E-08 14 56 0.31755 9.38E-07
50000 Case I 9 36 0.36461 8.84E-07 15 60 0.32342 8.14E-07
Case II 9 36 0.27196 4.12E-07 15 60 0.3605 5.66E-07
Case III 8 32 0.25501 1.18E-07 13 52 0.22338 2.87E-07
Case IV 9 36 0.22228 1.08E-07 16 64 0.26331 4.48E-07
Case V 10 40 0.25618 1.70E-07 16 64 0.27305 6.94E-07
Case VI 10 39 0.27278 1.87E-07 16 63 0.28175 5.01E-07
Case VII 10 40 0.22836 2.04E-07 15 60 0.75345 5.27E-07
100000 Case I 10 40 0.57002 1.03E-07 16 64 0.56535 2.89E-07
Case II 9 36 0.34715 5.83E-07 15 60 0.66238 8.01E-07
Case III 8 32 0.41892 1.67E-07 13 52 0.45765 4.06E-07
Case IV 9 36 0.34291 1.53E-07 16 64 0.68567 6.34E-07
Case V 10 40 0.53349 2.40E-07 16 64 0.51067 9.81E-07
Case VI 10 39 0.35763 2.64E-07 16 63 0.5497 7.09E-07
Case VII 10 40 0.55468 2.88E-07 15 60 0.6995 7.46E-07
INER. ALGO ALGO.
DIM INP NI NF CPU NORM NI NF CPU NORM
1000 Case I 9 36 0.029847 1.25E-07 14 56 0.010049 4.58E-07
Case II 8 32 0.01614 7.06E-07 14 56 0.009322 3.19E-07
Case III 7 28 0.036217 2.03E-07 11 44 0.014851 6.44E-07
Case IV 8 32 0.00896 1.86E-07 15 60 0.009756 2.52E-07
Case V 9 36 0.012299 2.91E-07 15 60 0.014388 3.91E-07
Case VI 9 35 0.012932 3.19E-07 15 59 0.01453 2.82E-07
Case VII 9 36 0.015981 3.43E-07 14 56 0.017579 3.00E-07
5000 Case I 9 36 0.040232 2.79E-07 15 60 0.06212 2.57E-07
Case II 9 36 0.038549 1.30E-07 14 56 0.035553 7.13E-07
Case III 7 28 0.02036 4.53E-07 12 48 0.037788 3.61E-07
Case IV 8 32 0.025654 4.15E-07 15 60 0.035425 5.64E-07
Case V 9 36 0.026546 6.51E-07 15 60 0.12771 8.73E-07
Case VI 9 35 0.096229 7.14E-07 15 59 0.044662 6.31E-07
Case VII 9 36 0.093227 7.84E-07 14 56 0.053641 6.71E-07
10000 Case I 9 36 0.054105 3.95E-07 15 60 0.055328 3.64E-07
Case II 9 36 0.10633 1.84E-07 15 60 0.10212 2.53E-07
Case III 7 28 0.037857 6.41E-07 12 48 0.066777 5.11E-07
Case IV 8 32 0.082416 5.87E-07 15 60 0.071481 7.98E-07
Case V 9 36 0.18582 9.20E-07 16 64 0.059125 3.10E-07
Case VI 10 39 0.070373 8.34E-08 15 59 0.081851 8.93E-07
Case VII 10 40 0.071963 9.12E-08 14 56 0.31755 9.38E-07
50000 Case I 9 36 0.36461 8.84E-07 15 60 0.32342 8.14E-07
Case II 9 36 0.27196 4.12E-07 15 60 0.3605 5.66E-07
Case III 8 32 0.25501 1.18E-07 13 52 0.22338 2.87E-07
Case IV 9 36 0.22228 1.08E-07 16 64 0.26331 4.48E-07
Case V 10 40 0.25618 1.70E-07 16 64 0.27305 6.94E-07
Case VI 10 39 0.27278 1.87E-07 16 63 0.28175 5.01E-07
Case VII 10 40 0.22836 2.04E-07 15 60 0.75345 5.27E-07
100000 Case I 10 40 0.57002 1.03E-07 16 64 0.56535 2.89E-07
Case II 9 36 0.34715 5.83E-07 15 60 0.66238 8.01E-07
Case III 8 32 0.41892 1.67E-07 13 52 0.45765 4.06E-07
Case IV 9 36 0.34291 1.53E-07 16 64 0.68567 6.34E-07
Case V 10 40 0.53349 2.40E-07 16 64 0.51067 9.81E-07
Case VI 10 39 0.35763 2.64E-07 16 63 0.5497 7.09E-07
Case VII 10 40 0.55468 2.88E-07 15 60 0.6995 7.46E-07
Table 9.  Test results for problem 8
INER. ALGO ALGO.
DIM INP NI NF CPU NORM NI NF CPU NORM
1000 Case I 11 38 0.032713 3.88E-07 16 58 0.00921 9.66E-07
Case II 11 38 0.010107 3.88E-07 16 58 0.00733 9.66E-07
Case III 11 39 0.0097 3.88E-07 16 59 0.010495 9.66E-07
Case IV 11 39 0.008542 3.88E-07 16 59 0.013631 9.66E-07
Case V 11 39 0.010247 3.88E-07 16 59 0.009248 9.66E-07
Case VI 11 39 0.014019 3.88E-07 16 59 0.01325 9.66E-07
Case VII 11 39 0.010456 3.88E-07 16 59 0.015507 9.66E-07
5000 Case I 8 29 0.029519 8.46E-07 12 46 0.03298 6.18E-07
Case II 8 30 0.028146 8.46E-07 12 46 0.028464 6.18E-07
Case III 8 30 0.031133 8.46E-07 12 46 0.026699 6.18E-07
Case IV 8 30 0.032266 8.46E-07 12 46 0.065305 6.18E-07
Case V 8 30 0.043717 8.46E-07 12 46 0.033667 6.18E-07
Case VI 8 30 0.075426 8.46E-07 12 46 0.032206 6.18E-07
Case VII 8 30 0.043783 8.46E-07 12 46 0.052994 6.18E-07
10000 Case I 7 26 0.070454 5.16E-07 9 35 0.045216 7.70E-07
Case II 7 27 0.053641 5.16E-07 9 35 0.059089 7.70E-07
Case III 7 27 0.094187 5.16E-07 9 35 0.045507 7.70E-07
Case IV 7 27 0.20473 5.16E-07 9 35 0.074929 7.70E-07
Case V 7 27 0.071282 5.16E-07 9 35 0.075532 7.70E-07
Case VI 7 27 0.064401 5.16E-07 9 35 0.065486 7.70E-07
Case VII 7 27 0.098105 5.16E-07 9 35 0.055328 7.70E-07
50000 Case I 6 24 0.22768 1.49E-07 12 48 0.35733 5.55E-07
Case II 6 24 0.28215 1.49E-07 12 48 0.34089 5.55E-07
Case III 6 24 0.72157 1.49E-07 12 48 0.30864 5.55E-07
Case IV 6 24 0.53086 1.49E-07 12 48 0.37903 5.55E-07
Case V 6 24 0.39711 1.49E-07 12 48 0.34614 5.55E-07
Case VI 6 24 0.48753 1.49E-07 12 48 0.41103 5.55E-07
Case VII 6 24 0.29061 1.49E-07 12 48 0.33605 5.55E-07
100000 Case I 6 24 0.66013 6.95E-07 6 24 0.43192 5.77E-07
Case II 6 24 0.60689 6.95E-07 6 24 0.30724 5.77E-07
Case III 6 24 0.78423 6.95E-07 6 24 0.46109 5.77E-07
Case IV 6 24 0.7826 6.95E-07 6 24 0.37262 5.77E-07
Case V 6 24 0.70311 6.95E-07 6 24 0.40775 5.77E-07
Case VI 6 24 0.9559 6.95E-07 6 24 0.38522 5.77E-07
Case VII 6 24 0.68726 6.95E-07 6 24 0.48134 5.77E-07
INER. ALGO ALGO.
DIM INP NI NF CPU NORM NI NF CPU NORM
1000 Case I 11 38 0.032713 3.88E-07 16 58 0.00921 9.66E-07
Case II 11 38 0.010107 3.88E-07 16 58 0.00733 9.66E-07
Case III 11 39 0.0097 3.88E-07 16 59 0.010495 9.66E-07
Case IV 11 39 0.008542 3.88E-07 16 59 0.013631 9.66E-07
Case V 11 39 0.010247 3.88E-07 16 59 0.009248 9.66E-07
Case VI 11 39 0.014019 3.88E-07 16 59 0.01325 9.66E-07
Case VII 11 39 0.010456 3.88E-07 16 59 0.015507 9.66E-07
5000 Case I 8 29 0.029519 8.46E-07 12 46 0.03298 6.18E-07
Case II 8 30 0.028146 8.46E-07 12 46 0.028464 6.18E-07
Case III 8 30 0.031133 8.46E-07 12 46 0.026699 6.18E-07
Case IV 8 30 0.032266 8.46E-07 12 46 0.065305 6.18E-07
Case V 8 30 0.043717 8.46E-07 12 46 0.033667 6.18E-07
Case VI 8 30 0.075426 8.46E-07 12 46 0.032206 6.18E-07
Case VII 8 30 0.043783 8.46E-07 12 46 0.052994 6.18E-07
10000 Case I 7 26 0.070454 5.16E-07 9 35 0.045216 7.70E-07
Case II 7 27 0.053641 5.16E-07 9 35 0.059089 7.70E-07
Case III 7 27 0.094187 5.16E-07 9 35 0.045507 7.70E-07
Case IV 7 27 0.20473 5.16E-07 9 35 0.074929 7.70E-07
Case V 7 27 0.071282 5.16E-07 9 35 0.075532 7.70E-07
Case VI 7 27 0.064401 5.16E-07 9 35 0.065486 7.70E-07
Case VII 7 27 0.098105 5.16E-07 9 35 0.055328 7.70E-07
50000 Case I 6 24 0.22768 1.49E-07 12 48 0.35733 5.55E-07
Case II 6 24 0.28215 1.49E-07 12 48 0.34089 5.55E-07
Case III 6 24 0.72157 1.49E-07 12 48 0.30864 5.55E-07
Case IV 6 24 0.53086 1.49E-07 12 48 0.37903 5.55E-07
Case V 6 24 0.39711 1.49E-07 12 48 0.34614 5.55E-07
Case VI 6 24 0.48753 1.49E-07 12 48 0.41103 5.55E-07
Case VII 6 24 0.29061 1.49E-07 12 48 0.33605 5.55E-07
100000 Case I 6 24 0.66013 6.95E-07 6 24 0.43192 5.77E-07
Case II 6 24 0.60689 6.95E-07 6 24 0.30724 5.77E-07
Case III 6 24 0.78423 6.95E-07 6 24 0.46109 5.77E-07
Case IV 6 24 0.7826 6.95E-07 6 24 0.37262 5.77E-07
Case V 6 24 0.70311 6.95E-07 6 24 0.40775 5.77E-07
Case VI 6 24 0.9559 6.95E-07 6 24 0.38522 5.77E-07
Case VII 6 24 0.68726 6.95E-07 6 24 0.48134 5.77E-07
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