American Institute of Mathematical Sciences

Advanced
January  2021, 17(1): 101-116. doi: 10.3934/jimo.2019101

A diagonal PRP-type projection method for convex constrained nonlinear monotone equations

Hassan Mohammad ,

Department of Mathematical Sciences, Faculty of Physical Sciences, Bayero University, Kano, Kano, 700241, Nigeria

* Corresponding author: Hassan Mohammad

Received  April 2018 Revised  May 2019 Published  January 2021 Early access  September 2019

Iterative methods for nonlinear monotone equations do not require the differentiability assumption on the residual function. This special property of the methods makes them suitable for solving large-scale nonsmooth monotone equations. In this work, we present a diagonal Polak-Ribi$ \grave{e} $re-Polyak (PRP) conjugate gradient-type method for solving large-scale nonlinear monotone equations with convex constraints. The search direction is a combine form of a multivariate (diagonal) spectral method and a modified PRP conjugate gradient method. Proper safeguards are devised to ensure positive definiteness of the diagonal matrix associated with the search direction. Based on Lipschitz continuity and monotonicity assumptions the method is shown to be globally convergent. Numerical results are presented by means of comparative experiments with recently proposed multivariate spectral Dai-Yuan-type (J. Ind. Manag. Optim. 13 (2017) 283-295) and Wei-Yao-Liu-type (Int. J. Comput. Math. 92 (2015) 2261-2272) conjugate gradient methods.

Keywords: Nonlinear monotone equations, diagonal spectral gradient method, PRP conjugate gradient method, global convergence.
Mathematics Subject Classification: Primary: 49M37, 65H10, 65K05; Secondary: 90C56.
Citation: Hassan Mohammad. A diagonal PRP-type projection method for convex constrained nonlinear monotone equations. Journal of Industrial and Management Optimization, 2021, 17 (1) : 101-116. doi: 10.3934/jimo.2019101
References:
[1]

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.

[2]

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

[3]

Y. Bing and G. Lin, An efficient implementation of Merrill's method for sparse or partially separable systems of nonlinear equations, SIAM J. Optim., 1 (1991), 206-221.  doi: 10.1137/0801015.

[4]

W. Cheng, A PRP type method for systems of monotone equations, Math. Comput. Modelling, 50 (2009), 15-20.  doi: 10.1016/j.mcm.2009.04.007.

[5]

Y. Dai and Y. Yuan, A nonlinear conjugate gradient method with a strong global convergence property, SIAM J. Optim., 10 (1999), 177-182.  doi: 10.1137/S1052623497318992.

[6]

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

[7]

X. L. Dong, H. Liu, Y. L. Xu and X. M. Yang, Some nonlinear conjugate gradient methods with sufficient descent condition and global convergence, Optim. Lett., 9 (2015), 1421-1432.  doi: 10.1007/s11590-014-0836-5.

[8]

M. Eshaghnezhad, S. Effati and A. Mansoori, A neurodynamic model to solve nonlinear pseudo-monotone projection equation and its applications, IEEE Transactions on Cybernetics, 47 (2017), 3050-3062.  doi: 10.1109/TCYB.2016.2611529.

[9]

M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Math. Programming, 53 (1992), 99-110.  doi: 10.1007/BF01585696.

[10]

B. Ghaddar, J. Marecek and M. Mevissen, Optimal power flow as a polynomial optimization problem, IEEE Transactions on Power Systems, 31 (2016), 539-546.  doi: 10.1109/TPWRS.2015.2390037.

[11]

B. Gu, V. S. Sheng, K. Y. Tay, W. Romano and S. Li, Incremental support vector learning for ordinal regression, IEEE Trans. Neural Netw. Learn. Syst., 26 (2015), 1403-1416.  doi: 10.1109/TNNLS.2014.2342533.

[12]

L. Han, G. Yu and L. Guan, Multivariate spectral gradient method for unconstrained optimization, Appl. Math. Comput., 201 (2008), 621-630.  doi: 10.1016/j.amc.2007.12.054.

[13]

Y. Hu and Z. Wei, Wei–Yao–Liu conjugate gradient projection algorithm for nonlinear monotone equations with convex constraints, Int. J. Comput. Math., 92 (2015), 2261-2272.  doi: 10.1080/00207160.2014.977879.

[14]

W. La Cruz, A projected derivative-free algorithm for nonlinear equations with convex constraints, Optim. Methods Softw., 29 (2014), 24-41.  doi: 10.1080/10556788.2012.721129.

[15]

W. La Cruz, A spectral algorithm for large-scale systems of nonlinear monotone equations, Numer. Algorithms, 76 (2017), 1109-1130.  doi: 10.1007/s11075-017-0299-8.

[16]

W. La Cruz, J. Martínez and M. Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Math. Comp., 75 (2006), 1429-1448.  doi: 10.1090/S0025-5718-06-01840-0.

[17]

J. Li, X. Li, B. Yang and X. Sun, Segmentation-based image copy-move forgery detection scheme, IEEE Transactions on Information Forensics and Security, 10 (2015), 507-518. 

[18]

Q. Li and D. H. Li, A class of derivative-free methods for large-scale nonlinear monotone equations, IMA J. Numer. Anal., 31 (2011), 1625-1635.  doi: 10.1093/imanum/drq015.

[19]

J. Liu and X. L. Du, A gradient projection method for the sparse signal reconstruction in compressive sensing, Appl. Anal., 97 (2018), 2122-2131.  doi: 10.1080/00036811.2017.1359556.

[20]

J. Liu and Y. Duan, Two spectral gradient projection methods for constrained equations and their linear convergence rate, J. Inequal. Appl., 2015 (2015), 13pp. doi: 10.1186/s13660-014-0525-z.

[21]

J. Liu and Y. Feng, A derivative-free iterative method for nonlinear monotone equations with convex constraints, Numerical Algorithms, 82 (2019), 1-18.  doi: 10.1007/s11075-018-0603-2.

[22]

J. Liu and S. Li, Multivariate spectral DY-type projection method for convex constrained nonlinear monotone equations, J. Ind. Manag. Optim., 13 (2017), 283-295.  doi: 10.3934/jimo.2016017.

[23]

J. Liu and S. Li, A projection method for convex constrained monotone nonlinear equations with applications, Comput. Math. Appl., 70 (2015), 2442-2453.  doi: 10.1016/j.camwa.2015.09.014.

[24]

S. Liu, Y. Huang and H. W. Jiao, Sufficient descent conjugate gradient methods for solving convex constrained nonlinear monotone equations, Abstr. Appl. Anal., 2014 (2014), 12pp. doi: 10.1155/2014/305643.

[25]

F. Ma and C. Wang, Modified projection method for solving a system of monotone equations with convex constraints, J. Appl. Math. Comput., 34 (2010), 47-56.  doi: 10.1007/s12190-009-0305-y.

[26]

H. Mohammad and A. B. Abubakar, A positive spectral gradient-like method for nonlinear monotone equations, Bull. Comput. Appl. Math., 5 (2017), 99-115. 

[27]

H. Mohammad and S. A. Santos, A structured diagonal Hessian approximation method with evaluation complexity analysis for nonlinear least squares, Comput. Appl. Math., 37 (2018), 6619-6653.  doi: 10.1007/s40314-018-0696-1.

[28]

Y. Ou and J. Li, A new derivative-free SCG-type projection method for nonlinear monotone equations with convex constraints, J. Appl. Math. Comput., 56 (2018), 195-216.  doi: 10.1007/s12190-016-1068-x.

[29]

Y. Ou and Y. Liu, Supermemory gradient methods for monotone nonlinear equations with convex constraints, Comput. Appl. Math., 36 (2017), 259-279.  doi: 10.1007/s40314-015-0228-1.

[30]

Z. Papp and S. Rapajić, FR type methods for systems of large-scale nonlinear monotone equations, Appl. Math. Comput., 269 (2015), 816-823.  doi: 10.1016/j.amc.2015.08.002.

[31]

E. Polak and G. Ribiere, Note sur la convergence de méthodes de directions conjuguées, Revue française d'informatique et de recherche opérationnelle. Série rouge, 3 (1969), 35–43.

[32]

B. T. Polyak, The conjugate gradient method in extremal problems, USSR Comp. Math. and Mathem. Physics, 9 (1969), 94-112.  doi: 10.1016/0041-5553(69)90035-4.

[33]

G. Qian, D. Han, L. Xu and H. Y., Solving nonadditive traffic assignment problems: A self-adaptive projection-auxiliary problem method for variational inequalities, J. Ind. Manag. Optim., 9 (2013), 255-274.  doi: 10.3934/jimo.2013.9.255.

[34]

M. V. Solodov and B. Svaiter, A globally convergent inexact Newton method for systems of monotone equations, in Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Applied Optimization, Springer, 1998,355–369. doi: 10.1007/978-1-4757-6388-1_18.

[35]

M. Sun and J. Liu, Three derivative-free projection methods for nonlinear equations with convex constraints, J. Appl. Math. Comput., 47 (2015), 265-276.  doi: 10.1007/s12190-014-0774-5.

[36]

C. Wang and Y. Wang, A superlinearly convergent projection method for constrained systems of nonlinear equations, J. Global Optim., 44 (2009), 283-296.  doi: 10.1007/s10898-008-9324-8.

[37]

C. Wang, Y. Wang and C. Xu, A projection method for a system of nonlinear monotone equations with convex constraints, Math. Methods Oper. Res., 66 (2007), 33-46.  doi: 10.1007/s00186-006-0140-y.

[38]

X. Wang, S. Li and X. Kou, A self-adaptive three-term conjugate gradient method for monotone nonlinear equations with convex constraints, Calcolo, 53 (2016), 133-145.  doi: 10.1007/s10092-015-0140-5.

[39]

A. J. Wood and B. F. Wollenberg, Power Generation, Operation, and Control, John Wiley & Sons, 2012.

[40]

Y. Xiao and H. Zhu, A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing, J. Math. Anal. Appl., 405 (2013), 310-319.  doi: 10.1016/j.jmaa.2013.04.017.

[41]

Q. Yan, X. Z. Peng and D. H. Li, A globally convergent derivative-free method for solving large-scale nonlinear monotone equations, J. Comput. Appl. Math., 234 (2010), 649-657.  doi: 10.1016/j.cam.2010.01.001.

[42]

G. Yu, S. Niu and J. Ma, Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints, J. Ind. Manag. Optim., 9 (2013), 117-129.  doi: 10.3934/jimo.2013.9.117.

[43]

Z. Yu, J. Lin, J. Sun, Y. H. Xiao, L. Liu and Z. H. Li, Spectral gradient projection method for monotone nonlinear equations with convex constraints, Appl. Numer. Math., 59 (2009), 2416-2423.  doi: 10.1016/j.apnum.2009.04.004.

[44]

N. Yuan, A derivative-free projection method for solving convex constrained monotone equations, SCIENCEASIA, 43 (2017), 195-200.  doi: 10.2306/scienceasia1513-1874.2017.43.195.

[45]

M. Zhang, Y. Xiao and H. Dou, Solving nonlinear constrained monotone equations via limited memory BFGS algorithm, J. of Comp. Infor. Syst., 7 (2011), 3995-4006. 

[46]

Y. Zheng, B. Jeon, D. Xu, Q. M. Wu and H. Zhang, Image segmentation by generalized hierarchical fuzzy c-means algorithm, J. of Int. & Fuzzy Syst., 28 (2015), 961-973. 

[47]

W. Zhou and D. H. Li, Limited memory BFGS method for nonlinear monotone equations, J. Comput. Math., 25 (2007), 89-96. 

[48]

W. Zhou and F. Wang, A PRP-based residual method for large-scale monotone nonlinear equations, Appl. Math. Comput., 261 (2015), 1-7.  doi: 10.1016/j.amc.2015.03.069.

show all references

References:
[1]

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.

[2]

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

[3]

Y. Bing and G. Lin, An efficient implementation of Merrill's method for sparse or partially separable systems of nonlinear equations, SIAM J. Optim., 1 (1991), 206-221.  doi: 10.1137/0801015.

[4]

W. Cheng, A PRP type method for systems of monotone equations, Math. Comput. Modelling, 50 (2009), 15-20.  doi: 10.1016/j.mcm.2009.04.007.

[5]

Y. Dai and Y. Yuan, A nonlinear conjugate gradient method with a strong global convergence property, SIAM J. Optim., 10 (1999), 177-182.  doi: 10.1137/S1052623497318992.

[6]

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

[7]

X. L. Dong, H. Liu, Y. L. Xu and X. M. Yang, Some nonlinear conjugate gradient methods with sufficient descent condition and global convergence, Optim. Lett., 9 (2015), 1421-1432.  doi: 10.1007/s11590-014-0836-5.

[8]

M. Eshaghnezhad, S. Effati and A. Mansoori, A neurodynamic model to solve nonlinear pseudo-monotone projection equation and its applications, IEEE Transactions on Cybernetics, 47 (2017), 3050-3062.  doi: 10.1109/TCYB.2016.2611529.

[9]

M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Math. Programming, 53 (1992), 99-110.  doi: 10.1007/BF01585696.

[10]

B. Ghaddar, J. Marecek and M. Mevissen, Optimal power flow as a polynomial optimization problem, IEEE Transactions on Power Systems, 31 (2016), 539-546.  doi: 10.1109/TPWRS.2015.2390037.

[11]

B. Gu, V. S. Sheng, K. Y. Tay, W. Romano and S. Li, Incremental support vector learning for ordinal regression, IEEE Trans. Neural Netw. Learn. Syst., 26 (2015), 1403-1416.  doi: 10.1109/TNNLS.2014.2342533.

[12]

L. Han, G. Yu and L. Guan, Multivariate spectral gradient method for unconstrained optimization, Appl. Math. Comput., 201 (2008), 621-630.  doi: 10.1016/j.amc.2007.12.054.

[13]

Y. Hu and Z. Wei, Wei–Yao–Liu conjugate gradient projection algorithm for nonlinear monotone equations with convex constraints, Int. J. Comput. Math., 92 (2015), 2261-2272.  doi: 10.1080/00207160.2014.977879.

[14]

W. La Cruz, A projected derivative-free algorithm for nonlinear equations with convex constraints, Optim. Methods Softw., 29 (2014), 24-41.  doi: 10.1080/10556788.2012.721129.

[15]

W. La Cruz, A spectral algorithm for large-scale systems of nonlinear monotone equations, Numer. Algorithms, 76 (2017), 1109-1130.  doi: 10.1007/s11075-017-0299-8.

[16]

W. La Cruz, J. Martínez and M. Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Math. Comp., 75 (2006), 1429-1448.  doi: 10.1090/S0025-5718-06-01840-0.

[17]

J. Li, X. Li, B. Yang and X. Sun, Segmentation-based image copy-move forgery detection scheme, IEEE Transactions on Information Forensics and Security, 10 (2015), 507-518. 

[18]

Q. Li and D. H. Li, A class of derivative-free methods for large-scale nonlinear monotone equations, IMA J. Numer. Anal., 31 (2011), 1625-1635.  doi: 10.1093/imanum/drq015.

[19]

J. Liu and X. L. Du, A gradient projection method for the sparse signal reconstruction in compressive sensing, Appl. Anal., 97 (2018), 2122-2131.  doi: 10.1080/00036811.2017.1359556.

[20]

J. Liu and Y. Duan, Two spectral gradient projection methods for constrained equations and their linear convergence rate, J. Inequal. Appl., 2015 (2015), 13pp. doi: 10.1186/s13660-014-0525-z.

[21]

J. Liu and Y. Feng, A derivative-free iterative method for nonlinear monotone equations with convex constraints, Numerical Algorithms, 82 (2019), 1-18.  doi: 10.1007/s11075-018-0603-2.

[22]

J. Liu and S. Li, Multivariate spectral DY-type projection method for convex constrained nonlinear monotone equations, J. Ind. Manag. Optim., 13 (2017), 283-295.  doi: 10.3934/jimo.2016017.

[23]

J. Liu and S. Li, A projection method for convex constrained monotone nonlinear equations with applications, Comput. Math. Appl., 70 (2015), 2442-2453.  doi: 10.1016/j.camwa.2015.09.014.

[24]

S. Liu, Y. Huang and H. W. Jiao, Sufficient descent conjugate gradient methods for solving convex constrained nonlinear monotone equations, Abstr. Appl. Anal., 2014 (2014), 12pp. doi: 10.1155/2014/305643.

[25]

F. Ma and C. Wang, Modified projection method for solving a system of monotone equations with convex constraints, J. Appl. Math. Comput., 34 (2010), 47-56.  doi: 10.1007/s12190-009-0305-y.

[26]

H. Mohammad and A. B. Abubakar, A positive spectral gradient-like method for nonlinear monotone equations, Bull. Comput. Appl. Math., 5 (2017), 99-115. 

[27]

H. Mohammad and S. A. Santos, A structured diagonal Hessian approximation method with evaluation complexity analysis for nonlinear least squares, Comput. Appl. Math., 37 (2018), 6619-6653.  doi: 10.1007/s40314-018-0696-1.

[28]

Y. Ou and J. Li, A new derivative-free SCG-type projection method for nonlinear monotone equations with convex constraints, J. Appl. Math. Comput., 56 (2018), 195-216.  doi: 10.1007/s12190-016-1068-x.

[29]

Y. Ou and Y. Liu, Supermemory gradient methods for monotone nonlinear equations with convex constraints, Comput. Appl. Math., 36 (2017), 259-279.  doi: 10.1007/s40314-015-0228-1.

[30]

Z. Papp and S. Rapajić, FR type methods for systems of large-scale nonlinear monotone equations, Appl. Math. Comput., 269 (2015), 816-823.  doi: 10.1016/j.amc.2015.08.002.

[31]

E. Polak and G. Ribiere, Note sur la convergence de méthodes de directions conjuguées, Revue française d'informatique et de recherche opérationnelle. Série rouge, 3 (1969), 35–43.

[32]

B. T. Polyak, The conjugate gradient method in extremal problems, USSR Comp. Math. and Mathem. Physics, 9 (1969), 94-112.  doi: 10.1016/0041-5553(69)90035-4.

[33]

G. Qian, D. Han, L. Xu and H. Y., Solving nonadditive traffic assignment problems: A self-adaptive projection-auxiliary problem method for variational inequalities, J. Ind. Manag. Optim., 9 (2013), 255-274.  doi: 10.3934/jimo.2013.9.255.

[34]

M. V. Solodov and B. Svaiter, A globally convergent inexact Newton method for systems of monotone equations, in Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Applied Optimization, Springer, 1998,355–369. doi: 10.1007/978-1-4757-6388-1_18.

[35]

M. Sun and J. Liu, Three derivative-free projection methods for nonlinear equations with convex constraints, J. Appl. Math. Comput., 47 (2015), 265-276.  doi: 10.1007/s12190-014-0774-5.

[36]

C. Wang and Y. Wang, A superlinearly convergent projection method for constrained systems of nonlinear equations, J. Global Optim., 44 (2009), 283-296.  doi: 10.1007/s10898-008-9324-8.

[37]

C. Wang, Y. Wang and C. Xu, A projection method for a system of nonlinear monotone equations with convex constraints, Math. Methods Oper. Res., 66 (2007), 33-46.  doi: 10.1007/s00186-006-0140-y.

[38]

X. Wang, S. Li and X. Kou, A self-adaptive three-term conjugate gradient method for monotone nonlinear equations with convex constraints, Calcolo, 53 (2016), 133-145.  doi: 10.1007/s10092-015-0140-5.

[39]

A. J. Wood and B. F. Wollenberg, Power Generation, Operation, and Control, John Wiley & Sons, 2012.

[40]

Y. Xiao and H. Zhu, A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing, J. Math. Anal. Appl., 405 (2013), 310-319.  doi: 10.1016/j.jmaa.2013.04.017.

[41]

Q. Yan, X. Z. Peng and D. H. Li, A globally convergent derivative-free method for solving large-scale nonlinear monotone equations, J. Comput. Appl. Math., 234 (2010), 649-657.  doi: 10.1016/j.cam.2010.01.001.

[42]

G. Yu, S. Niu and J. Ma, Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints, J. Ind. Manag. Optim., 9 (2013), 117-129.  doi: 10.3934/jimo.2013.9.117.

[43]

Z. Yu, J. Lin, J. Sun, Y. H. Xiao, L. Liu and Z. H. Li, Spectral gradient projection method for monotone nonlinear equations with convex constraints, Appl. Numer. Math., 59 (2009), 2416-2423.  doi: 10.1016/j.apnum.2009.04.004.

[44]

N. Yuan, A derivative-free projection method for solving convex constrained monotone equations, SCIENCEASIA, 43 (2017), 195-200.  doi: 10.2306/scienceasia1513-1874.2017.43.195.

[45]

M. Zhang, Y. Xiao and H. Dou, Solving nonlinear constrained monotone equations via limited memory BFGS algorithm, J. of Comp. Infor. Syst., 7 (2011), 3995-4006. 

[46]

Y. Zheng, B. Jeon, D. Xu, Q. M. Wu and H. Zhang, Image segmentation by generalized hierarchical fuzzy c-means algorithm, J. of Int. & Fuzzy Syst., 28 (2015), 961-973. 

[47]

W. Zhou and D. H. Li, Limited memory BFGS method for nonlinear monotone equations, J. Comput. Math., 25 (2007), 89-96. 

[48]

W. Zhou and F. Wang, A PRP-based residual method for large-scale monotone nonlinear equations, Appl. Math. Comput., 261 (2015), 1-7.  doi: 10.1016/j.amc.2015.03.069.

Figure 1.  Performance profile with respect to number of iterations (ITER)
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Figure 2.  Performance profile with respect to number of function evaluations
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Figure 3.  Performance profile with respect to CPU time
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Table 1.  The initial points used for the test problems
INITIAL POINT VALUE
$ x_1 $ $ (1, 1, \ldots , 1)^T $
$ x_2 $ $ (0.1, 0.1, \ldots , 0.1)^T $
$ x_3 $ $ \bigl(\frac{1}{2}, \frac{1}{2^2}, \ldots , \frac{1}{2^n}\bigr)^T $
$ x_4 $ $ \bigl(0, 1-\frac{1}{2}, \ldots , 1-\frac{1}{n}\bigr)^T $
$ x_5 $ $ \bigl(0, \frac{1}{n}, \ldots , \frac{n-1}{n}\bigr)^T $
$ x_6 $ $ \bigl(1, \frac{1}{2}, \ldots , \frac{1}{n}\bigr)^T $
$ x_7 $ $ \bigl(n-\frac{1}{n}, n- \frac{2}{n}, \ldots , n-1 \bigr)^T $
$ x_8 $ $ \bigl(\frac{1}{n}, \frac{2}{n}, \ldots , 1\bigr)^T $
Table Options

Download as excel

INITIAL POINT VALUE
$ x_1 $ $ (1, 1, \ldots , 1)^T $
$ x_2 $ $ (0.1, 0.1, \ldots , 0.1)^T $
$ x_3 $ $ \bigl(\frac{1}{2}, \frac{1}{2^2}, \ldots , \frac{1}{2^n}\bigr)^T $
$ x_4 $ $ \bigl(0, 1-\frac{1}{2}, \ldots , 1-\frac{1}{n}\bigr)^T $
$ x_5 $ $ \bigl(0, \frac{1}{n}, \ldots , \frac{n-1}{n}\bigr)^T $
$ x_6 $ $ \bigl(1, \frac{1}{2}, \ldots , \frac{1}{n}\bigr)^T $
$ x_7 $ $ \bigl(n-\frac{1}{n}, n- \frac{2}{n}, \ldots , n-1 \bigr)^T $
$ x_8 $ $ \bigl(\frac{1}{n}, \frac{2}{n}, \ldots , 1\bigr)^T $
Table 2.  Numerical Results for DPPM, MDYP and WYLP for Problem 1 with given initial points and dimension, $ f $ represents failure
DPPM MDYP WYLP
DIMENSION INITIAL POINT ITER FVAL TIME NORM ITER FVAL TIME NORM ITER FVAL TIME NORM
1000 $ x_1 $ 5 14 0.2355 8.79E-08 16 69 0.0200 4.84E-06 2 9 0.0085 0.00E+00
$ x_2 $ 4 11 0.0156 1.06E-08 10 40 0.0126 5.44E-06 4 15 0.0079 0.00E+00
$ x_3 $ 15 35 0.0887 6.01E-06 13 54 0.1723 7.55E-06 5 20 0.0119 0.00E+00
$ x_4 $ 6 16 0.0204 5.47E-06 25 133 0.1284 3.35E-06 2 9 0.0079 0.00E+00
$ x_5 $ 8 20 0.0294 2.47E-06 27 138 0.0757 6.03E-06 2 11 0.0076 0.00E+00
$ x_6 $ 8 19 0.0362 1.95E-06 26 134 0.0350 4.52E-06 4 15 0.0102 0.00E+00
$ x_7 $ 8 20 0.0292 2.48E-06 25 133 0.1248 3.35E-06 2 11 0.0071 0.00E+00
$ x_8 $ 8 20 0.0299 2.49E-06 22 112 0.0317 4.53E-06 2 11 0.0080 0.00E+00
$ x_1 $ 5 14 0.0720 1.71E-07 17 73 0.0820 2.67E-06 2 9 0.0143 0.00E+00
$ x_2 $ 4 11 0.0457 2.31E-08 8 31 0.0563 8.85E-06 4 15 0.0182 0.00E+00
$ x_3 $ 15 35 0.0707 6.05E-06 13 54 1.5146 7.55E-06 5 20 0.0308 0.00E+00
$ x_4 $ 6 16 0.0492 5.61E-06 24 126 0.1178 2.59E-06 2 9 0.0154 0.00E+00
$ x_5 $ 8 20 0.0611 5.54E-06 39 226 0.4130 3.08E-06 2 11 0.0190 0.00E+00
$ x_6 $ 8 19 0.0831 1.95E-06 22 99 0.1434 5.73E-06 4 15 0.0251 0.00E+00
$ x_7 $ 8 20 0.0651 5.54E-06 24 126 0.1656 2.59E-06 2 11 0.0175 0.00E+00
$ x_8 $ 8 20 0.0476 5.55E-06 35 216 0.3063 8.62E-06 2 11 0.0242 0.00E+00
$ x_1 $ 5 14 0.0956 2.37E-07 15 60 0.1038 4.70E-06 2 9 0.0256 0.00E+00
$ x_2 $ 4 11 0.0494 3.25E-08 9 34 0.0697 5.29E-06 4 15 0.0457 0.00E+00
$ x_3 $ 15 35 0.2145 6.05E-06 13 54 3.1315 7.55E-06 5 20 0.0614 0.00E+00
$ x_4 $ 6 16 0.0723 5.75E-06 52 336 0.7500 6.45E-06 2 9 0.0262 0.00E+00
$ x_5 $ 8 20 0.1135 7.84E-06 39 257 0.8268 7.40E-06 2 11 0.0234 0.00E+00
$ x_6 $ 8 19 0.1167 1.95E-06 21 97 0.1880 7.11E-06 4 15 0.0360 0.00E+00
$ x_7 $ 8 20 0.1115 7.84E-06 58 456 0.8633 8.74E-06 2 11 0.0253 0.00E+00
$ x_8 $ 8 20 0.1302 7.84E-06 40 250 0.3797 4.87E-06 2 11 0.0324 0.00E+00
$ x_1 $ 7 23 0.3971 9.16E-11 14 56 0.2812 4.32E-06 3 16 0.1857 0.00E+00
$ x_2 $ 4 11 0.1609 7.25E-08 10 38 0.3246 3.24E-06 4 15 0.1384 0.00E+00
$ x_3 $ 15 35 0.6173 6.05E-06 13 54 25.8148 7.55E-06 5 20 0.1416 0.00E+00
$ x_4 $ 7 23 0.4199 7.98E-10 41 287 2.6315 4.61E-06 2 13 0.1230 0.00E+00
$ x_5 $ 9 23 0.3295 2.21E-06 43 261 7.0197 8.15E-06 2 10 0.0809 0.00E+00
$ x_6 $ 8 19 0.4273 1.95E-06 23 121 0.3488 2.56E-06 4 15 0.1403 0.00E+00
$ x_7 $ 9 23 0.4588 2.21E-06 42 295 2.2382 7.02E-06 3 13 0.1398 0.00E+00
$ x_8 $ 9 23 0.3960 2.21E-06 41 252 2.0126 5.39E-07 3 13 0.1062 0.00E+00
$ x_1 $ 7 27 0.9775 3.55E-07 14 56 0.5772 4.93E-06 3 18 0.5132 0.00E+00
$ x_2 $ 4 11 0.4880 1.03E-07 10 38 0.5872 4.73E-06 4 15 0.2445 0.00E+00
$ x_3 $ 15 35 1.0573 6.05E-06 13 54 87.3280 7.55E-06 5 20 0.3240 0.00E+00
$ x_4 $ 7 27 0.5609 3.63E-06 41 265 3.9792 2.29E-06 3 18 0.4003 0.00E+00
$ x_5 $ 9 24 1.1510 4.99E-06 52 420 13.2180 1.53E-06 2 10 0.2150 0.00E+00
$ x_6 $ 11 25 0.7559 9.20E-08 21 99 1.2739 4.64E-06 4 15 0.2824 0.00E+00
$ x_7 $ 9 24 1.0529 4.99E-06 41 266 3.6878 6.90E-06 3 13 0.1655 0.00E+00
$ x_8 $ 9 24 0.8091 4.99E-06 42 316 15.6351 3.58E-06 3 13 0.2459 0.00E+00
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DPPM MDYP WYLP
DIMENSION INITIAL POINT ITER FVAL TIME NORM ITER FVAL TIME NORM ITER FVAL TIME NORM
1000 $ x_1 $ 5 14 0.2355 8.79E-08 16 69 0.0200 4.84E-06 2 9 0.0085 0.00E+00
$ x_2 $ 4 11 0.0156 1.06E-08 10 40 0.0126 5.44E-06 4 15 0.0079 0.00E+00
$ x_3 $ 15 35 0.0887 6.01E-06 13 54 0.1723 7.55E-06 5 20 0.0119 0.00E+00
$ x_4 $ 6 16 0.0204 5.47E-06 25 133 0.1284 3.35E-06 2 9 0.0079 0.00E+00
$ x_5 $ 8 20 0.0294 2.47E-06 27 138 0.0757 6.03E-06 2 11 0.0076 0.00E+00
$ x_6 $ 8 19 0.0362 1.95E-06 26 134 0.0350 4.52E-06 4 15 0.0102 0.00E+00
$ x_7 $ 8 20 0.0292 2.48E-06 25 133 0.1248 3.35E-06 2 11 0.0071 0.00E+00
$ x_8 $ 8 20 0.0299 2.49E-06 22 112 0.0317 4.53E-06 2 11 0.0080 0.00E+00
$ x_1 $ 5 14 0.0720 1.71E-07 17 73 0.0820 2.67E-06 2 9 0.0143 0.00E+00
$ x_2 $ 4 11 0.0457 2.31E-08 8 31 0.0563 8.85E-06 4 15 0.0182 0.00E+00
$ x_3 $ 15 35 0.0707 6.05E-06 13 54 1.5146 7.55E-06 5 20 0.0308 0.00E+00
$ x_4 $ 6 16 0.0492 5.61E-06 24 126 0.1178 2.59E-06 2 9 0.0154 0.00E+00
$ x_5 $ 8 20 0.0611 5.54E-06 39 226 0.4130 3.08E-06 2 11 0.0190 0.00E+00
$ x_6 $ 8 19 0.0831 1.95E-06 22 99 0.1434 5.73E-06 4 15 0.0251 0.00E+00
$ x_7 $ 8 20 0.0651 5.54E-06 24 126 0.1656 2.59E-06 2 11 0.0175 0.00E+00
$ x_8 $ 8 20 0.0476 5.55E-06 35 216 0.3063 8.62E-06 2 11 0.0242 0.00E+00
$ x_1 $ 5 14 0.0956 2.37E-07 15 60 0.1038 4.70E-06 2 9 0.0256 0.00E+00
$ x_2 $ 4 11 0.0494 3.25E-08 9 34 0.0697 5.29E-06 4 15 0.0457 0.00E+00
$ x_3 $ 15 35 0.2145 6.05E-06 13 54 3.1315 7.55E-06 5 20 0.0614 0.00E+00
$ x_4 $ 6 16 0.0723 5.75E-06 52 336 0.7500 6.45E-06 2 9 0.0262 0.00E+00
$ x_5 $ 8 20 0.1135 7.84E-06 39 257 0.8268 7.40E-06 2 11 0.0234 0.00E+00
$ x_6 $ 8 19 0.1167 1.95E-06 21 97 0.1880 7.11E-06 4 15 0.0360 0.00E+00
$ x_7 $ 8 20 0.1115 7.84E-06 58 456 0.8633 8.74E-06 2 11 0.0253 0.00E+00
$ x_8 $ 8 20 0.1302 7.84E-06 40 250 0.3797 4.87E-06 2 11 0.0324 0.00E+00
$ x_1 $ 7 23 0.3971 9.16E-11 14 56 0.2812 4.32E-06 3 16 0.1857 0.00E+00
$ x_2 $ 4 11 0.1609 7.25E-08 10 38 0.3246 3.24E-06 4 15 0.1384 0.00E+00
$ x_3 $ 15 35 0.6173 6.05E-06 13 54 25.8148 7.55E-06 5 20 0.1416 0.00E+00
$ x_4 $ 7 23 0.4199 7.98E-10 41 287 2.6315 4.61E-06 2 13 0.1230 0.00E+00
$ x_5 $ 9 23 0.3295 2.21E-06 43 261 7.0197 8.15E-06 2 10 0.0809 0.00E+00
$ x_6 $ 8 19 0.4273 1.95E-06 23 121 0.3488 2.56E-06 4 15 0.1403 0.00E+00
$ x_7 $ 9 23 0.4588 2.21E-06 42 295 2.2382 7.02E-06 3 13 0.1398 0.00E+00
$ x_8 $ 9 23 0.3960 2.21E-06 41 252 2.0126 5.39E-07 3 13 0.1062 0.00E+00
$ x_1 $ 7 27 0.9775 3.55E-07 14 56 0.5772 4.93E-06 3 18 0.5132 0.00E+00
$ x_2 $ 4 11 0.4880 1.03E-07 10 38 0.5872 4.73E-06 4 15 0.2445 0.00E+00
$ x_3 $ 15 35 1.0573 6.05E-06 13 54 87.3280 7.55E-06 5 20 0.3240 0.00E+00
$ x_4 $ 7 27 0.5609 3.63E-06 41 265 3.9792 2.29E-06 3 18 0.4003 0.00E+00
$ x_5 $ 9 24 1.1510 4.99E-06 52 420 13.2180 1.53E-06 2 10 0.2150 0.00E+00
$ x_6 $ 11 25 0.7559 9.20E-08 21 99 1.2739 4.64E-06 4 15 0.2824 0.00E+00
$ x_7 $ 9 24 1.0529 4.99E-06 41 266 3.6878 6.90E-06 3 13 0.1655 0.00E+00
$ x_8 $ 9 24 0.8091 4.99E-06 42 316 15.6351 3.58E-06 3 13 0.2459 0.00E+00
Table 3.  Numerical Results for DPPM, MDYP and WYLP for Problem 2 with given initial points and dimension, $ f $ represents failure
DPPM MDYP WYLP
DIMENSION INITIAL POINT ITER FVAL TIME NORM ITER FVAL TIME NORM ITER FVAL TIME NORM
1000 $ x_1 $ 9 26 0.0452 4.37E-06 13 40 0.0794 3.39E-06 19 93 0.3206 9.75E-07
$ x_2 $ 7 19 0.0244 7.69E-06 7 22 0.0300 5.13E-06 15 73 0.0934 4.17E-07
$ x_3 $ 8 22 0.0703 3.51E-06 11 36 0.1053 3.48E-06 15 73 0.1950 4.96E-07
$ x_4 $ 9 26 0.0401 6.71E-06 14 46 0.0217 7.56E-06 19 93 0.0969 9.62E-07
$ x_5 $ 10 29 0.0461 3.13E-06 14 46 0.0238 7.56E-06 19 93 0.1020 4.80E-07
$ x_6 $ 9 25 0.0377 6.09E-06 10 32 0.0135 8.15E-06 16 78 0.0441 6.75E-07
$ x_7 $ 10 29 0.0764 3.13E-06 14 46 0.0165 7.56E-06 19 93 0.0604 4.80E-07
$ x_8 $ 10 29 0.0419 3.13E-06 14 46 0.0264 7.53E-06 19 93 0.0568 4.81E-07
$ x_1 $ 9 26 0.1909 9.84E-06 13 40 0.0641 7.50E-06 20 98 0.4133 8.51E-07
$ x_2 $ 8 22 0.0677 3.39E-06 8 25 0.0360 4.94E-06 15 73 0.1504 9.03E-07
$ x_3 $ 8 22 0.1055 3.51E-06 11 36 1.2062 8.57E-06 15 73 0.1302 4.89E-07
$ x_4 $ 10 29 0.1083 2.25E-06 35 157 0.1346 7.40E-06 20 98 0.1267 8.49E-07
$ x_5 $ 10 29 0.1217 7.01E-06 31 138 0.4007 4.24E-06 20 98 0.1494 4.20E-07
$ x_6 $ 9 25 0.0721 6.02E-06 10 32 0.0487 8.13E-06 16 78 0.1823 6.64E-07
$ x_7 $ 10 29 0.1134 7.01E-06 31 138 0.2469 4.24E-06 20 98 0.1852 4.20E-07
$ x_8 $ 10 29 0.0780 7.01E-06 22 88 0.3009 7.25E-06 20 98 0.1700 4.21E-07
$ x_1 $ 10 29 0.1738 2.79E-06 14 43 0.1073 2.89E-06 21 103 0.5148 4.80E-07
$ x_2 $ 8 22 0.0853 4.78E-06 8 25 0.0799 6.92E-06 16 78 0.2554 5.09E-07
$ x_3 $ 8 22 0.0803 3.51E-06 11 36 3.0274 9.89E-06 15 73 0.1792 4.88E-07
$ x_4 $ 10 29 0.1861 2.99E-06 16 54 0.4752 4.83E-06 21 103 0.3229 4.79E-07
$ x_5 $ 10 29 0.2444 9.91E-06 16 54 0.1676 4.83E-06 20 98 0.3376 5.93E-07
$ x_6 $ 9 25 0.2425 6.02E-06 10 32 0.0720 8.13E-06 16 78 0.2158 6.63E-07
$ x_7 $ 10 29 0.2000 9.91E-06 16 54 0.1314 4.83E-06 20 98 0.2923 5.93E-07
$ x_8 $ 10 29 0.2440 9.92E-06 16 54 0.0871 4.69E-06 20 98 0.3689 5.93E-07
$ x_1 $ 12 39 1.0114 2.30E-06 14 43 0.4292 6.46E-06 23 116 2.6259 6.55E-07
$ x_2 $ 9 25 0.6191 2.13E-06 9 28 0.3785 1.05E-06 17 83 0.9079 4.53E-07
$ x_3 $ 8 22 0.3316 3.51E-06 12 39 24.3728 3.64E-06 15 73 0.4278 4.88E-07
$ x_4 $ 12 39 0.6189 2.30E-06 f f f f 23 116 1.4482 6.55E-07
$ x_5 $ 11 32 0.8394 4.43E-06 f f f f 21 103 1.7160 5.29E-07
$ x_6 $ 9 25 0.5946 6.01E-06 10 32 1.1865 8.12E-06 16 78 0.7065 6.62E-07
$ x_7 $ 11 32 0.7496 4.43E-06 f f f f 21 103 1.1586 5.29E-07
$ x_8 $ 11 32 0.8790 4.44E-06 f f f f 21 103 0.8051 5.29E-07
$ x_1 $ 12 45 2.1717 2.90E-06 14 43 1.1046 9.13E-06 25 130 3.6530 6.38E-07
$ x_2 $ 9 25 1.3327 3.02E-06 9 28 0.7541 1.49E-06 17 81 1.5444 1.60E-11
$ x_3 $ 8 22 0.9087 3.51E-06 12 41 94.8845 3.66E-06 15 73 1.2575 4.88E-07
$ x_4 $ 12 45 1.7092 3.05E-06 f f f f 25 130 1.9217 6.38E-07
$ x_5 $ 13 40 1.2000 2.81E-06 f f f f 22 110 2.0242 6.94E-07
$ x_6 $ 9 25 1.2705 6.01E-06 10 32 0.8082 8.12E-06 16 78 1.4797 6.62E-07
$ x_7 $ 13 40 1.9988 2.81E-06 f f f f 22 110 1.7161 6.94E-07
$ x_8 $ 13 40 1.7253 2.81E-06 f f f f 22 110 2.3099 6.94E-07
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DPPM MDYP WYLP
DIMENSION INITIAL POINT ITER FVAL TIME NORM ITER FVAL TIME NORM ITER FVAL TIME NORM
1000 $ x_1 $ 9 26 0.0452 4.37E-06 13 40 0.0794 3.39E-06 19 93 0.3206 9.75E-07
$ x_2 $ 7 19 0.0244 7.69E-06 7 22 0.0300 5.13E-06 15 73 0.0934 4.17E-07
$ x_3 $ 8 22 0.0703 3.51E-06 11 36 0.1053 3.48E-06 15 73 0.1950 4.96E-07
$ x_4 $ 9 26 0.0401 6.71E-06 14 46 0.0217 7.56E-06 19 93 0.0969 9.62E-07
$ x_5 $ 10 29 0.0461 3.13E-06 14 46 0.0238 7.56E-06 19 93 0.1020 4.80E-07
$ x_6 $ 9 25 0.0377 6.09E-06 10 32 0.0135 8.15E-06 16 78 0.0441 6.75E-07
$ x_7 $ 10 29 0.0764 3.13E-06 14 46 0.0165 7.56E-06 19 93 0.0604 4.80E-07
$ x_8 $ 10 29 0.0419 3.13E-06 14 46 0.0264 7.53E-06 19 93 0.0568 4.81E-07
$ x_1 $ 9 26 0.1909 9.84E-06 13 40 0.0641 7.50E-06 20 98 0.4133 8.51E-07
$ x_2 $ 8 22 0.0677 3.39E-06 8 25 0.0360 4.94E-06 15 73 0.1504 9.03E-07
$ x_3 $ 8 22 0.1055 3.51E-06 11 36 1.2062 8.57E-06 15 73 0.1302 4.89E-07
$ x_4 $ 10 29 0.1083 2.25E-06 35 157 0.1346 7.40E-06 20 98 0.1267 8.49E-07
$ x_5 $ 10 29 0.1217 7.01E-06 31 138 0.4007 4.24E-06 20 98 0.1494 4.20E-07
$ x_6 $ 9 25 0.0721 6.02E-06 10 32 0.0487 8.13E-06 16 78 0.1823 6.64E-07
$ x_7 $ 10 29 0.1134 7.01E-06 31 138 0.2469 4.24E-06 20 98 0.1852 4.20E-07
$ x_8 $ 10 29 0.0780 7.01E-06 22 88 0.3009 7.25E-06 20 98 0.1700 4.21E-07
$ x_1 $ 10 29 0.1738 2.79E-06 14 43 0.1073 2.89E-06 21 103 0.5148 4.80E-07
$ x_2 $ 8 22 0.0853 4.78E-06 8 25 0.0799 6.92E-06 16 78 0.2554 5.09E-07
$ x_3 $ 8 22 0.0803 3.51E-06 11 36 3.0274 9.89E-06 15 73 0.1792 4.88E-07
$ x_4 $ 10 29 0.1861 2.99E-06 16 54 0.4752 4.83E-06 21 103 0.3229 4.79E-07
$ x_5 $ 10 29 0.2444 9.91E-06 16 54 0.1676 4.83E-06 20 98 0.3376 5.93E-07
$ x_6 $ 9 25 0.2425 6.02E-06 10 32 0.0720 8.13E-06 16 78 0.2158 6.63E-07
$ x_7 $ 10 29 0.2000 9.91E-06 16 54 0.1314 4.83E-06 20 98 0.2923 5.93E-07
$ x_8 $ 10 29 0.2440 9.92E-06 16 54 0.0871 4.69E-06 20 98 0.3689 5.93E-07
$ x_1 $ 12 39 1.0114 2.30E-06 14 43 0.4292 6.46E-06 23 116 2.6259 6.55E-07
$ x_2 $ 9 25 0.6191 2.13E-06 9 28 0.3785 1.05E-06 17 83 0.9079 4.53E-07
$ x_3 $ 8 22 0.3316 3.51E-06 12 39 24.3728 3.64E-06 15 73 0.4278 4.88E-07
$ x_4 $ 12 39 0.6189 2.30E-06 f f f f 23 116 1.4482 6.55E-07
$ x_5 $ 11 32 0.8394 4.43E-06 f f f f 21 103 1.7160 5.29E-07
$ x_6 $ 9 25 0.5946 6.01E-06 10 32 1.1865 8.12E-06 16 78 0.7065 6.62E-07
$ x_7 $ 11 32 0.7496 4.43E-06 f f f f 21 103 1.1586 5.29E-07
$ x_8 $ 11 32 0.8790 4.44E-06 f f f f 21 103 0.8051 5.29E-07
$ x_1 $ 12 45 2.1717 2.90E-06 14 43 1.1046 9.13E-06 25 130 3.6530 6.38E-07
$ x_2 $ 9 25 1.3327 3.02E-06 9 28 0.7541 1.49E-06 17 81 1.5444 1.60E-11
$ x_3 $ 8 22 0.9087 3.51E-06 12 41 94.8845 3.66E-06 15 73 1.2575 4.88E-07
$ x_4 $ 12 45 1.7092 3.05E-06 f f f f 25 130 1.9217 6.38E-07
$ x_5 $ 13 40 1.2000 2.81E-06 f f f f 22 110 2.0242 6.94E-07
$ x_6 $ 9 25 1.2705 6.01E-06 10 32 0.8082 8.12E-06 16 78 1.4797 6.62E-07
$ x_7 $ 13 40 1.9988 2.81E-06 f f f f 22 110 1.7161 6.94E-07
$ x_8 $ 13 40 1.7253 2.81E-06 f f f f 22 110 2.3099 6.94E-07
Table 4.  Numerical Results for DPPM, MDYP and WYLP for Problem 3 with given initial points and dimension, $ f $ represents failure
DPPM MDYP WYLP
DIMENSION INITIAL POINT ITER FVAL TIME NORM ITER FVAL TIME NORM ITER FVAL TIME NORM
1000 $ x_1 $ 4 8 0.8195 6.47E-09 12 39 0.0871 9.82E-06 2 8 0.0309 0.00E+00
$ x_2 $ 3 6 0.0912 8.25E-08 10 32 0.0564 9.57E-06 2 8 0.0068 0.00E+00
$ x_3 $ 13 27 0.0953 8.81E-06 9 29 0.2011 2.35E-06 13 51 0.0271 3.85E-07
$ x_4 $ 4 8 0.0192 1.42E-06 37 210 0.1077 2.98E-06 17 69 0.0250 7.21E-07
$ x_5 $ 6 12 0.0705 5.56E-06 37 210 0.0689 2.98E-06 17 73 0.0373 8.83E-07
$ x_6 $ 5 10 0.0919 6.68E-11 16 56 0.0211 7.32E-06 15 59 0.0292 8.78E-07
$ x_7 $ 6 12 0.0203 5.56E-06 37 210 0.0639 2.98E-06 17 73 0.0332 8.83E-07
$ x_8 $ 8 16 0.1217 4.35E-08 19 85 0.0470 6.92E-06 17 73 0.0263 8.83E-07
$ x_1 $ 4 8 0.0746 1.45E-08 13 42 0.0529 3.54E-06 2 8 0.0133 0.00E+00
$ x_2 $ 3 6 0.0249 1.84E-07 11 35 0.0574 3.71E-06 2 8 0.0097 0.00E+00
$ x_3 $ 13 27 0.1299 8.81E-06 9 29 1.1781 2.35E-06 13 51 0.0781 3.85E-07
$ x_4 $ 4 8 0.0397 1.28E-06 36 209 0.4912 3.49E-06 18 72 0.0891 3.15E-07
$ x_5 $ 13 26 0.0749 7.27E-13 36 209 0.3499 3.49E-06 18 76 0.1789 6.06E-07
$ x_6 $ 5 10 0.0320 6.70E-11 21 81 0.1014 7.63E-06 16 62 0.0643 9.17E-07
$ x_7 $ 13 26 0.0643 7.27E-13 36 209 0.3509 3.49E-06 18 76 0.1192 6.08E-07
$ x_8 $ 9 18 0.0524 8.51E-09 20 84 0.1209 8.44E-06 18 76 0.0923 6.07E-07
$ x_1 $ 4 8 0.3372 2.04E-08 13 42 0.0841 5.00E-06 2 8 0.0186 0.00E+00
$ x_2 $ 3 6 0.0404 2.61E-07 11 35 0.0801 5.24E-06 2 8 0.0165 0.00E+00
$ x_3 $ 13 27 0.1017 8.81E-06 9 29 3.2921 2.35E-06 13 51 0.0892 3.85E-07
$ x_4 $ 4 8 0.0452 1.30E-06 31 177 0.4489 1.82E-06 15 59 0.0737 8.90E-07
$ x_5 $ 11 22 0.1231 1.43E-10 31 177 0.3576 1.82E-06 18 76 0.0882 8.57E-07
$ x_6 $ 5 10 0.0408 6.70E-11 18 72 0.1468 8.19E-06 f f f f
$ x_7 $ 11 22 0.0840 1.43E-10 31 177 0.3460 1.82E-06 18 76 0.1498 8.57E-07
$ x_8 $ 12 24 0.0726 1.21E-11 29 153 0.3801 6.17E-06 18 76 0.1678 8.55E-07
$ x_1 $ 5 15 0.1876 1.58E-06 14 45 0.3743 4.52E-06 3 14 0.2255 0.00E+00
$ x_2 $ 3 6 0.1362 5.83E-07 12 38 0.3290 4.63E-06 2 8 0.1014 0.00E+00
$ x_3 $ 13 27 0.4447 8.81E-06 9 29 22.4652 2.35E-06 13 51 0.3770 3.85E-07
$ x_4 $ 6 17 0.3568 2.93E-10 40 278 8.4671 7.10E-06 18 73 0.6564 6.79E-07
$ x_5 $ 17 36 0.6444 7.27E-06 40 278 9.2475 7.10E-06 f f f f
$ x_6 $ 5 10 0.1576 6.71E-11 21 80 4.8421 9.53E-06 f f f f
$ x_7 $ 17 36 0.4210 7.27E-06 40 278 8.9672 7.10E-06 f f f f
$ x_8 $ 17 36 0.4061 7.27E-06 36 222 0.7206 4.06E-06 f f f f
$ x_1 $ 6 21 0.9043 8.13E-09 14 45 0.3008 6.39E-06 4 21 0.2761 0.00E+00
$ x_2 $ 3 6 0.2392 8.25E-07 12 38 0.6761 6.55E-06 2 8 0.1008 0.00E+00
$ x_3 $ 13 27 0.5491 8.81E-06 9 29 72.1280 2.35E-06 13 51 0.5589 3.85E-07
$ x_4 $ 6 21 0.4396 1.81E-08 34 224 3.5081 4.61E-06 19 80 1.1189 6.33E-07
$ x_5 $ 18 40 1.4053 7.38E-06 34 224 2.4146 4.61E-06 f f f f
$ x_6 $ 5 10 0.3521 6.71E-11 19 71 7.7563 4.00E-06 f f f f
$ x_7 $ 18 40 1.0043 7.38E-06 34 224 3.4369 4.61E-06 f f f f
$ x_8 $ 18 40 1.2596 7.38E-06 37 225 3.3577 4.58E-06 f f f f
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DPPM MDYP WYLP
DIMENSION INITIAL POINT ITER FVAL TIME NORM ITER FVAL TIME NORM ITER FVAL TIME NORM
1000 $ x_1 $ 4 8 0.8195 6.47E-09 12 39 0.0871 9.82E-06 2 8 0.0309 0.00E+00
$ x_2 $ 3 6 0.0912 8.25E-08 10 32 0.0564 9.57E-06 2 8 0.0068 0.00E+00
$ x_3 $ 13 27 0.0953 8.81E-06 9 29 0.2011 2.35E-06 13 51 0.0271 3.85E-07
$ x_4 $ 4 8 0.0192 1.42E-06 37 210 0.1077 2.98E-06 17 69 0.0250 7.21E-07
$ x_5 $ 6 12 0.0705 5.56E-06 37 210 0.0689 2.98E-06 17 73 0.0373 8.83E-07
$ x_6 $ 5 10 0.0919 6.68E-11 16 56 0.0211 7.32E-06 15 59 0.0292 8.78E-07
$ x_7 $ 6 12 0.0203 5.56E-06 37 210 0.0639 2.98E-06 17 73 0.0332 8.83E-07
$ x_8 $ 8 16 0.1217 4.35E-08 19 85 0.0470 6.92E-06 17 73 0.0263 8.83E-07
$ x_1 $ 4 8 0.0746 1.45E-08 13 42 0.0529 3.54E-06 2 8 0.0133 0.00E+00
$ x_2 $ 3 6 0.0249 1.84E-07 11 35 0.0574 3.71E-06 2 8 0.0097 0.00E+00
$ x_3 $ 13 27 0.1299 8.81E-06 9 29 1.1781 2.35E-06 13 51 0.0781 3.85E-07
$ x_4 $ 4 8 0.0397 1.28E-06 36 209 0.4912 3.49E-06 18 72 0.0891 3.15E-07
$ x_5 $ 13 26 0.0749 7.27E-13 36 209 0.3499 3.49E-06 18 76 0.1789 6.06E-07
$ x_6 $ 5 10 0.0320 6.70E-11 21 81 0.1014 7.63E-06 16 62 0.0643 9.17E-07
$ x_7 $ 13 26 0.0643 7.27E-13 36 209 0.3509 3.49E-06 18 76 0.1192 6.08E-07
$ x_8 $ 9 18 0.0524 8.51E-09 20 84 0.1209 8.44E-06 18 76 0.0923 6.07E-07
$ x_1 $ 4 8 0.3372 2.04E-08 13 42 0.0841 5.00E-06 2 8 0.0186 0.00E+00
$ x_2 $ 3 6 0.0404 2.61E-07 11 35 0.0801 5.24E-06 2 8 0.0165 0.00E+00
$ x_3 $ 13 27 0.1017 8.81E-06 9 29 3.2921 2.35E-06 13 51 0.0892 3.85E-07
$ x_4 $ 4 8 0.0452 1.30E-06 31 177 0.4489 1.82E-06 15 59 0.0737 8.90E-07
$ x_5 $ 11 22 0.1231 1.43E-10 31 177 0.3576 1.82E-06 18 76 0.0882 8.57E-07
$ x_6 $ 5 10 0.0408 6.70E-11 18 72 0.1468 8.19E-06 f f f f
$ x_7 $ 11 22 0.0840 1.43E-10 31 177 0.3460 1.82E-06 18 76 0.1498 8.57E-07
$ x_8 $ 12 24 0.0726 1.21E-11 29 153 0.3801 6.17E-06 18 76 0.1678 8.55E-07
$ x_1 $ 5 15 0.1876 1.58E-06 14 45 0.3743 4.52E-06 3 14 0.2255 0.00E+00
$ x_2 $ 3 6 0.1362 5.83E-07 12 38 0.3290 4.63E-06 2 8 0.1014 0.00E+00
$ x_3 $ 13 27 0.4447 8.81E-06 9 29 22.4652 2.35E-06 13 51 0.3770 3.85E-07
$ x_4 $ 6 17 0.3568 2.93E-10 40 278 8.4671 7.10E-06 18 73 0.6564 6.79E-07
$ x_5 $ 17 36 0.6444 7.27E-06 40 278 9.2475 7.10E-06 f f f f
$ x_6 $ 5 10 0.1576 6.71E-11 21 80 4.8421 9.53E-06 f f f f
$ x_7 $ 17 36 0.4210 7.27E-06 40 278 8.9672 7.10E-06 f f f f
$ x_8 $ 17 36 0.4061 7.27E-06 36 222 0.7206 4.06E-06 f f f f
$ x_1 $ 6 21 0.9043 8.13E-09 14 45 0.3008 6.39E-06 4 21 0.2761 0.00E+00
$ x_2 $ 3 6 0.2392 8.25E-07 12 38 0.6761 6.55E-06 2 8 0.1008 0.00E+00
$ x_3 $ 13 27 0.5491 8.81E-06 9 29 72.1280 2.35E-06 13 51 0.5589 3.85E-07
$ x_4 $ 6 21 0.4396 1.81E-08 34 224 3.5081 4.61E-06 19 80 1.1189 6.33E-07
$ x_5 $ 18 40 1.4053 7.38E-06 34 224 2.4146 4.61E-06 f f f f
$ x_6 $ 5 10 0.3521 6.71E-11 19 71 7.7563 4.00E-06 f f f f
$ x_7 $ 18 40 1.0043 7.38E-06 34 224 3.4369 4.61E-06 f f f f
$ x_8 $ 18 40 1.2596 7.38E-06 37 225 3.3577 4.58E-06 f f f f
Table 5.  Numerical Results for DPPM, MDYP and WYLP for Problem 4 with given initial points and dimension, $ f $ represents failure
DPPM MDYP WYLP
DIMENSION INITIAL POINT ITER FVAL TIME NORM ITER FVAL TIME NORM ITER FVAL TIME NORM
1000 $ x_1 $ 10 29 0.1975 5.56E-06 13 41 0.0277 2.04E-06 17 83 0.0574 5.78E-07
$ x_2 $ 10 29 0.0528 8.48E-06 13 41 0.0191 3.11E-06 19 89 0.0732 7.34E-07
$ x_3 $ 10 28 0.0530 1.66E-14 13 41 0.0182 3.22E-06 f f f f
$ x_4 $ 10 28 0.0521 1.31E-14 13 41 0.0229 2.65E-06 f f f f
$ x_5 $ 10 28 0.0362 1.23E-14 13 41 0.0206 2.65E-06 20 92 0.0418 6.37E-07
$ x_6 $ 10 28 0.0370 1.12E-14 13 41 0.0291 3.22E-06 17 83 0.0544 7.43E-07
$ x_7 $ 10 28 0.0345 1.23E-14 13 41 0.0273 2.65E-06 20 92 0.0623 6.37E-07
$ x_8 $ 10 28 0.0388 1.16E-14 13 41 0.0181 2.65E-06 20 92 0.0504 6.37E-07
$ x_1 $ 11 32 0.1438 2.49E-06 13 41 0.2237 4.56E-06 f f f f
$ x_2 $ 10 30 0.1277 0.00E+00 13 41 0.2464 6.96E-06 657 2006 2.2801 4.71E-07
$ x_3 $ 11 34 0.1592 9.60E-06 13 41 0.0676 7.22E-06 21 104 0.2050 3.40E-08
$ x_4 $ 11 32 0.1546 2.49E-06 13 41 0.0814 5.94E-06 17 83 0.1518 8.10E-07
$ x_5 $ 11 34 0.0902 7.90E-06 13 41 0.1253 5.94E-06 21 105 0.1327 6.96E-07
$ x_6 $ 11 34 0.2370 9.60E-06 13 41 0.0804 7.22E-06 19 94 0.1391 1.91E-07
$ x_7 $ 11 34 0.1372 7.90E-06 13 41 0.0817 5.94E-06 21 105 0.1985 6.96E-07
$ x_8 $ 11 34 0.1759 7.90E-06 13 41 0.0618 5.94E-06 21 105 0.1921 6.96E-07
$ x_1 $ 10 30 0.3047 8.88E-16 13 41 0.1117 6.46E-06 248 777 2.4396 7.71E-07
$ x_2 $ 12 41 0.4791 8.80E-06 13 41 0.2053 9.84E-06 321 999 3.1750 7.52E-07
$ x_3 $ 11 37 0.2859 1.33E-15 14 44 0.2424 4.02E-06 f f f f
$ x_4 $ 11 34 0.2346 8.59E-06 13 41 0.1677 8.41E-06 27 116 0.1964 4.43E-07
$ x_5 $ 12 39 0.3669 4.87E-06 13 41 0.1120 8.41E-06 22 111 0.4084 6.30E-07
$ x_6 $ 12 41 0.2340 9.13E-06 14 44 0.1076 4.02E-06 240 756 2.1246 6.20E-07
$ x_7 $ 12 39 0.3153 4.87E-06 13 41 0.1737 8.41E-06 22 111 0.5055 6.30E-07
$ x_8 $ 12 39 0.3989 4.87E-06 13 41 0.1488 8.40E-06 22 111 0.3394 6.30E-07
$ x_1 $ 9 38 0.8360 0.00E+00 14 44 0.6011 5.69E-06 22 107 1.0331 4.71E-07
$ x_2 $ 13 66 1.4044 9.93E-14 14 44 0.7478 8.67E-06 23 121 1.4131 8.76E-07
$ x_3 $ 13 69 1.1444 9.93E-16 14 44 0.7075 9.00E-06 29 140 1.7453 9.92E-07
$ x_4 $ 10 41 1.0490 2.03E-14 14 44 0.5255 7.40E-06 f f f f
$ x_5 $ 11 53 1.4307 4.86E-14 14 44 0.6108 7.40E-06 277 878 5.8666 8.74E-07
$ x_6 $ 14 72 1.6761 7.15E-15 14 44 0.6519 9.00E-06 f f f f
$ x_7 $ 11 53 1.0887 4.86E-14 14 44 0.5013 7.40E-06 277 878 5.8247 8.74E-07
$ x_8 $ 11 53 0.9404 4.86E-14 14 44 0.6483 7.40E-06 23 116 1.6068 8.75E-07
$ x_1 $ 13 61 2.9900 1.40E-13 14 44 1.2238 8.04E-06 21 114 2.6706 4.32E-07
$ x_2 $ 15 92 4.1893 1.40E-13 15 47 1.0116 2.14E-06 24 140 3.1769 7.35E-07
$ x_3 $ 15 98 3.8911 1.40E-13 15 47 1.2457 2.23E-06 29 158 2.5033 8.14E-07
$ x_4 $ 11 55 2.2741 1.72E-14 15 47 1.3337 1.83E-06 21 114 1.5601 6.03E-07
$ x_5 $ 13 77 3.0852 9.15E-14 15 47 1.4839 1.83E-06 75 284 7.6840 8.99E-07
$ x_6 $ 15 98 3.7006 1.33E-13 15 47 1.2636 2.23E-06 567 1772 31.9028 7.18E-07
$ x_7 $ 13 77 3.3259 9.15E-14 15 47 1.3969 1.83E-06 75 284 5.1793 8.99E-07
$ x_8 $ 13 77 2.3221 9.15E-14 15 47 1.4001 1.83E-06 75 284 5.6734 8.99E-07
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DPPM MDYP WYLP
DIMENSION INITIAL POINT ITER FVAL TIME NORM ITER FVAL TIME NORM ITER FVAL TIME NORM
1000 $ x_1 $ 10 29 0.1975 5.56E-06 13 41 0.0277 2.04E-06 17 83 0.0574 5.78E-07
$ x_2 $ 10 29 0.0528 8.48E-06 13 41 0.0191 3.11E-06 19 89 0.0732 7.34E-07
$ x_3 $ 10 28 0.0530 1.66E-14 13 41 0.0182 3.22E-06 f f f f
$ x_4 $ 10 28 0.0521 1.31E-14 13 41 0.0229 2.65E-06 f f f f
$ x_5 $ 10 28 0.0362 1.23E-14 13 41 0.0206 2.65E-06 20 92 0.0418 6.37E-07
$ x_6 $ 10 28 0.0370 1.12E-14 13 41 0.0291 3.22E-06 17 83 0.0544 7.43E-07
$ x_7 $ 10 28 0.0345 1.23E-14 13 41 0.0273 2.65E-06 20 92 0.0623 6.37E-07
$ x_8 $ 10 28 0.0388 1.16E-14 13 41 0.0181 2.65E-06 20 92 0.0504 6.37E-07
$ x_1 $ 11 32 0.1438 2.49E-06 13 41 0.2237 4.56E-06 f f f f
$ x_2 $ 10 30 0.1277 0.00E+00 13 41 0.2464 6.96E-06 657 2006 2.2801 4.71E-07
$ x_3 $ 11 34 0.1592 9.60E-06 13 41 0.0676 7.22E-06 21 104 0.2050 3.40E-08
$ x_4 $ 11 32 0.1546 2.49E-06 13 41 0.0814 5.94E-06 17 83 0.1518 8.10E-07
$ x_5 $ 11 34 0.0902 7.90E-06 13 41 0.1253 5.94E-06 21 105 0.1327 6.96E-07
$ x_6 $ 11 34 0.2370 9.60E-06 13 41 0.0804 7.22E-06 19 94 0.1391 1.91E-07
$ x_7 $ 11 34 0.1372 7.90E-06 13 41 0.0817 5.94E-06 21 105 0.1985 6.96E-07
$ x_8 $ 11 34 0.1759 7.90E-06 13 41 0.0618 5.94E-06 21 105 0.1921 6.96E-07
$ x_1 $ 10 30 0.3047 8.88E-16 13 41 0.1117 6.46E-06 248 777 2.4396 7.71E-07
$ x_2 $ 12 41 0.4791 8.80E-06 13 41 0.2053 9.84E-06 321 999 3.1750 7.52E-07
$ x_3 $ 11 37 0.2859 1.33E-15 14 44 0.2424 4.02E-06 f f f f
$ x_4 $ 11 34 0.2346 8.59E-06 13 41 0.1677 8.41E-06 27 116 0.1964 4.43E-07
$ x_5 $ 12 39 0.3669 4.87E-06 13 41 0.1120 8.41E-06 22 111 0.4084 6.30E-07
$ x_6 $ 12 41 0.2340 9.13E-06 14 44 0.1076 4.02E-06 240 756 2.1246 6.20E-07
$ x_7 $ 12 39 0.3153 4.87E-06 13 41 0.1737 8.41E-06 22 111 0.5055 6.30E-07
$ x_8 $ 12 39 0.3989 4.87E-06 13 41 0.1488 8.40E-06 22 111 0.3394 6.30E-07
$ x_1 $ 9 38 0.8360 0.00E+00 14 44 0.6011 5.69E-06 22 107 1.0331 4.71E-07
$ x_2 $ 13 66 1.4044 9.93E-14 14 44 0.7478 8.67E-06 23 121 1.4131 8.76E-07
$ x_3 $ 13 69 1.1444 9.93E-16 14 44 0.7075 9.00E-06 29 140 1.7453 9.92E-07
$ x_4 $ 10 41 1.0490 2.03E-14 14 44 0.5255 7.40E-06 f f f f
$ x_5 $ 11 53 1.4307 4.86E-14 14 44 0.6108 7.40E-06 277 878 5.8666 8.74E-07
$ x_6 $ 14 72 1.6761 7.15E-15 14 44 0.6519 9.00E-06 f f f f
$ x_7 $ 11 53 1.0887 4.86E-14 14 44 0.5013 7.40E-06 277 878 5.8247 8.74E-07
$ x_8 $ 11 53 0.9404 4.86E-14 14 44 0.6483 7.40E-06 23 116 1.6068 8.75E-07
$ x_1 $ 13 61 2.9900 1.40E-13 14 44 1.2238 8.04E-06 21 114 2.6706 4.32E-07
$ x_2 $ 15 92 4.1893 1.40E-13 15 47 1.0116 2.14E-06 24 140 3.1769 7.35E-07
$ x_3 $ 15 98 3.8911 1.40E-13 15 47 1.2457 2.23E-06 29 158 2.5033 8.14E-07
$ x_4 $ 11 55 2.2741 1.72E-14 15 47 1.3337 1.83E-06 21 114 1.5601 6.03E-07
$ x_5 $ 13 77 3.0852 9.15E-14 15 47 1.4839 1.83E-06 75 284 7.6840 8.99E-07
$ x_6 $ 15 98 3.7006 1.33E-13 15 47 1.2636 2.23E-06 567 1772 31.9028 7.18E-07
$ x_7 $ 13 77 3.3259 9.15E-14 15 47 1.3969 1.83E-06 75 284 5.1793 8.99E-07
$ x_8 $ 13 77 2.3221 9.15E-14 15 47 1.4001 1.83E-06 75 284 5.6734 8.99E-07
Table 6.  Numerical Results for DPPM, MDYP and WYLP for Problem 5 with given initial points and dimension, $ f $ represents failure
DPPM MDYP WYLP
DIMENSION INITIAL POINT ITER FVAL TIME NORM ITER FVAL TIME NORM ITER FVAL TIME NORM
1000 $ x_1 $ 5 12 0.2244 3.26E-06 11 38 0.2095 3.21E-06 2 9 0.0260 0.00E+00
$ x_2 $ 4 8 0.1030 7.57E-08 9 29 0.0294 8.71E-06 2 8 0.0143 0.00E+00
$ x_3 $ 5 11 0.0745 1.28E-07 12 38 0.1401 1.70E-06 4 14 0.0322 2.22E-16
$ x_4 $ 7 16 0.0238 1.92E-06 21 79 0.0319 6.66E-06 12 45 0.0305 7.78E-08
$ x_5 $ 9 19 0.0954 6.02E-06 21 79 0.0333 6.66E-06 14 54 0.0385 3.41E-07
$ x_6 $ 9 19 0.0316 4.09E-06 11 36 0.0131 3.32E-06 14 54 0.0249 2.67E-07
$ x_7 $ 9 19 0.0215 6.02E-06 21 79 0.0265 6.66E-06 14 54 0.0204 3.42E-07
$ x_8 $ 9 19 0.0286 5.97E-06 17 61 0.0164 6.86E-06 14 54 0.0198 3.92E-07
$ x_1 $ 5 12 0.0353 7.29E-06 11 38 0.0340 7.19E-06 2 9 0.0136 0.00E+00
$ x_2 $ 4 8 0.0247 1.69E-07 10 32 0.0334 8.33E-06 2 8 0.0197 0.00E+00
$ x_3 $ 5 11 0.0330 1.28E-07 12 38 1.4578 1.70E-06 4 14 0.0172 2.22E-16
$ x_4 $ 7 16 0.0494 1.79E-06 19 69 0.0617 2.22E-06 12 47 0.0369 9.52E-07
$ x_5 $ 9 20 0.0565 2.04E-06 19 69 0.0724 2.22E-06 15 59 0.0548 3.37E-07
$ x_6 $ 9 19 0.0677 4.92E-06 11 36 0.0424 3.32E-06 14 54 0.0385 6.25E-07
$ x_7 $ 9 20 0.0583 2.04E-06 19 69 0.0908 2.22E-06 15 59 0.0642 3.32E-07
$ x_8 $ 9 20 0.0440 2.04E-06 27 122 0.1471 3.72E-06 15 59 0.0605 3.36E-07
$ x_1 $ 6 14 0.0693 3.67E-10 12 41 0.0766 4.72E-06 2 9 0.0312 0.00E+00
$ x_2 $ 4 8 0.0665 2.39E-07 11 35 0.0800 1.49E-06 2 8 0.0131 0.00E+00
$ x_3 $ 5 11 0.0537 1.28E-07 12 38 2.5950 1.70E-06 4 14 0.0239 2.22E-16
$ x_4 $ 8 18 0.1273 3.07E-07 17 62 0.1395 9.13E-06 12 47 0.0824 7.20E-07
$ x_5 $ 9 20 0.0858 2.89E-06 17 62 0.1122 9.13E-06 15 59 0.0614 4.78E-07
$ x_6 $ 9 19 0.1316 4.92E-06 11 36 0.0749 3.32E-06 14 54 0.0617 7.36E-07
$ x_7 $ 9 20 0.0844 2.89E-06 17 62 0.1038 9.13E-06 15 59 0.0613 4.75E-07
$ x_8 $ 9 20 0.1005 2.89E-06 27 103 0.4661 2.52E-06 15 59 0.0524 4.75E-07
$ x_1 $ 7 20 0.4618 3.27E-08 13 44 0.3079 7.07E-07 2 10 0.0474 0.00E+00
$ x_2 $ 4 8 0.1123 5.35E-07 11 35 0.3012 3.33E-06 2 8 0.0923 0.00E+00
$ x_3 $ 5 11 0.1031 1.28E-07 12 38 23.8270 1.70E-06 4 14 0.1142 2.22E-16
$ x_4 $ 7 20 0.4022 3.43E-08 18 65 0.5567 8.62E-06 10 42 0.2322 7.89E-08
$ x_5 $ 9 21 0.3592 4.85E-06 18 65 0.3972 8.62E-06 16 65 0.4585 8.55E-07
$ x_6 $ 9 19 0.3959 4.92E-06 11 36 0.2964 3.32E-06 f f f f
$ x_7 $ 9 21 0.3742 4.85E-06 18 65 0.4319 8.62E-06 16 65 0.1987 8.49E-07
$ x_8 $ 9 21 0.3703 4.85E-06 18 65 0.5524 8.97E-06 16 65 0.5407 8.39E-07
$ x_1 $ 7 23 0.7369 4.77E-06 13 44 0.7618 1.00E-06 3 16 0.1932 0.00E+00
$ x_2 $ 4 8 0.2658 7.57E-07 11 35 0.4434 4.71E-06 2 8 0.1158 0.00E+00
$ x_3 $ 5 11 0.3353 1.28E-07 12 38 107.4606 1.70E-06 4 14 0.2615 2.22E-16
$ x_4 $ 7 23 0.7591 4.81E-06 19 68 0.4613 2.35E-06 14 59 0.6242 5.49E-07
$ x_5 $ 9 22 0.7743 8.65E-06 19 68 1.1533 2.35E-06 f f f f
$ x_6 $ 9 19 0.6254 4.92E-06 11 36 0.5491 3.32E-06 f f f f
$ x_7 $ 9 22 0.7055 8.65E-06 19 68 1.0239 2.35E-06 f f f f
$ x_8 $ 9 22 0.5172 8.66E-06 19 68 0.9601 2.41E-06 f f f f
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DPPM MDYP WYLP
DIMENSION INITIAL POINT ITER FVAL TIME NORM ITER FVAL TIME NORM ITER FVAL TIME NORM
1000 $ x_1 $ 5 12 0.2244 3.26E-06 11 38 0.2095 3.21E-06 2 9 0.0260 0.00E+00
$ x_2 $ 4 8 0.1030 7.57E-08 9 29 0.0294 8.71E-06 2 8 0.0143 0.00E+00
$ x_3 $ 5 11 0.0745 1.28E-07 12 38 0.1401 1.70E-06 4 14 0.0322 2.22E-16
$ x_4 $ 7 16 0.0238 1.92E-06 21 79 0.0319 6.66E-06 12 45 0.0305 7.78E-08
$ x_5 $ 9 19 0.0954 6.02E-06 21 79 0.0333 6.66E-06 14 54 0.0385 3.41E-07
$ x_6 $ 9 19 0.0316 4.09E-06 11 36 0.0131 3.32E-06 14 54 0.0249 2.67E-07
$ x_7 $ 9 19 0.0215 6.02E-06 21 79 0.0265 6.66E-06 14 54 0.0204 3.42E-07
$ x_8 $ 9 19 0.0286 5.97E-06 17 61 0.0164 6.86E-06 14 54 0.0198 3.92E-07
$ x_1 $ 5 12 0.0353 7.29E-06 11 38 0.0340 7.19E-06 2 9 0.0136 0.00E+00
$ x_2 $ 4 8 0.0247 1.69E-07 10 32 0.0334 8.33E-06 2 8 0.0197 0.00E+00
$ x_3 $ 5 11 0.0330 1.28E-07 12 38 1.4578 1.70E-06 4 14 0.0172 2.22E-16
$ x_4 $ 7 16 0.0494 1.79E-06 19 69 0.0617 2.22E-06 12 47 0.0369 9.52E-07
$ x_5 $ 9 20 0.0565 2.04E-06 19 69 0.0724 2.22E-06 15 59 0.0548 3.37E-07
$ x_6 $ 9 19 0.0677 4.92E-06 11 36 0.0424 3.32E-06 14 54 0.0385 6.25E-07
$ x_7 $ 9 20 0.0583 2.04E-06 19 69 0.0908 2.22E-06 15 59 0.0642 3.32E-07
$ x_8 $ 9 20 0.0440 2.04E-06 27 122 0.1471 3.72E-06 15 59 0.0605 3.36E-07
$ x_1 $ 6 14 0.0693 3.67E-10 12 41 0.0766 4.72E-06 2 9 0.0312 0.00E+00
$ x_2 $ 4 8 0.0665 2.39E-07 11 35 0.0800 1.49E-06 2 8 0.0131 0.00E+00
$ x_3 $ 5 11 0.0537 1.28E-07 12 38 2.5950 1.70E-06 4 14 0.0239 2.22E-16
$ x_4 $ 8 18 0.1273 3.07E-07 17 62 0.1395 9.13E-06 12 47 0.0824 7.20E-07
$ x_5 $ 9 20 0.0858 2.89E-06 17 62 0.1122 9.13E-06 15 59 0.0614 4.78E-07
$ x_6 $ 9 19 0.1316 4.92E-06 11 36 0.0749 3.32E-06 14 54 0.0617 7.36E-07
$ x_7 $ 9 20 0.0844 2.89E-06 17 62 0.1038 9.13E-06 15 59 0.0613 4.75E-07
$ x_8 $ 9 20 0.1005 2.89E-06 27 103 0.4661 2.52E-06 15 59 0.0524 4.75E-07
$ x_1 $ 7 20 0.4618 3.27E-08 13 44 0.3079 7.07E-07 2 10 0.0474 0.00E+00
$ x_2 $ 4 8 0.1123 5.35E-07 11 35 0.3012 3.33E-06 2 8 0.0923 0.00E+00
$ x_3 $ 5 11 0.1031 1.28E-07 12 38 23.8270 1.70E-06 4 14 0.1142 2.22E-16
$ x_4 $ 7 20 0.4022 3.43E-08 18 65 0.5567 8.62E-06 10 42 0.2322 7.89E-08
$ x_5 $ 9 21 0.3592 4.85E-06 18 65 0.3972 8.62E-06 16 65 0.4585 8.55E-07
$ x_6 $ 9 19 0.3959 4.92E-06 11 36 0.2964 3.32E-06 f f f f
$ x_7 $ 9 21 0.3742 4.85E-06 18 65 0.4319 8.62E-06 16 65 0.1987 8.49E-07
$ x_8 $ 9 21 0.3703 4.85E-06 18 65 0.5524 8.97E-06 16 65 0.5407 8.39E-07
$ x_1 $ 7 23 0.7369 4.77E-06 13 44 0.7618 1.00E-06 3 16 0.1932 0.00E+00
$ x_2 $ 4 8 0.2658 7.57E-07 11 35 0.4434 4.71E-06 2 8 0.1158 0.00E+00
$ x_3 $ 5 11 0.3353 1.28E-07 12 38 107.4606 1.70E-06 4 14 0.2615 2.22E-16
$ x_4 $ 7 23 0.7591 4.81E-06 19 68 0.4613 2.35E-06 14 59 0.6242 5.49E-07
$ x_5 $ 9 22 0.7743 8.65E-06 19 68 1.1533 2.35E-06 f f f f
$ x_6 $ 9 19 0.6254 4.92E-06 11 36 0.5491 3.32E-06 f f f f
$ x_7 $ 9 22 0.7055 8.65E-06 19 68 1.0239 2.35E-06 f f f f
$ x_8 $ 9 22 0.5172 8.66E-06 19 68 0.9601 2.41E-06 f f f f
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